Moduli Spaces of Anti-Invariant Vector Bundles Over Curves and Conformal Blocks Hacen Zelaci

Moduli spaces of anti-invariant vector bundles over curves and conformal blocks Hacen Zelaci

To cite this version:

Hacen Zelaci. Moduli spaces of anti-invariant vector bundles over curves and conformal blocks. General Mathematics [math.GM]. Université Côte d’Azur, 2017. English. NNT : 2017AZUR4063. tel- 01679267

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Th`ese de doctorat

Pr´esent´eeen vue de l’obtention du grade de docteur en Mathematiques´ de l’Universite´ Coteˆ d’Azur

Par Hacen ZELACI

Espaces de Modules de Fibr´esVectoriels Anti-invariants sur les Courbes et Blocs Conformes

Dirig´eepar Pr. Christian PAULY

Soutenue le 29 Septembre 2017. Devant le jury compos´ede:

M. Arnaud BEAUVILLE Professeur, UniversitéCôted’Azur, LJAD Examinateur M. Indranil BISWAS Professeur, Tata Institute of Fundamental Research Rapporteur M. Jochen HEINLOTH Professeur, UniversitéDuisburg-Essen Rapporteur M. Christian PAULY Professeur, UniversitéCôted’Azur, LJAD Directeur M. Carlos SIMPSON Directeur de recherche CNRS, UniversitéCôted’Azur, LJAD Examinateur M. Christoph SORGER Professeur, Universitéde Nantes Examinateur

Moduli Spaces of Anti-invariant Vector Bundles over Curves and Conformal Blocks

August 22, 2017

iii

Abstract

by Hacen ZELACI

Moduli Spaces of Anti-invariant Vector Bundles over Curves and Conformal Blocks Let X be a smooth irreducible projective curve with an involution σ. In this dissertation, we study the moduli spaces of invariant and anti-invariant vector bundles over X under the induced action of σ. We introduce the notion of σ−quadratic modules and use it, with GIT, to construct these moduli spaces, and than we study some of their main properties. It turn out that these moduli spaces correspond to moduli spaces of parahoric G−torsors on the quotient curve X/σ, for some parahoric Bruhat-Tits group schemes G, which are twisted in the anti-invariant case. We study the Hitchin system over these moduli spaces and use it to derive a classification of their connected components using dominant maps from Prym varieties. We also study the determinant of cohomology line bundle on the moduli spaces of anti-invariant vector bundles. In some cases this line bundle admits some square roots called Pfaffian of cohomology line bundles. We prove that the spaces of global sections of the powers of these line bundles (spaces of generalized theta functions) can be canonically identified with the conformal blocks for some twisted affine Kac- Moody Lie algebras of type A(2).

Résumé Espaces de Modules des FibrésVectoriels Anti-invariants sur les Courbes et Blocs Conformes.

Soit X une courbe projective lisse et irréductiblemunie d’une involution σ. Dans cette thèse,nous étudionsles fibrésvectoriels invariants and anti-invariants sur X sous l’action induite par σ. On introduit la notion de modules σ−quadratiques et on l’utilise, avec GIT, pour construire ces espaces de modules, puis on en étudie certaines propriétés.Ces espaces de modules correspondent aux espaces de modules de G−torseurs parahoriques sur la courbe X/σ, pour certains schémasen groupes parahoriques G de type Bruhat-Tits, qui sont twistésdans le cas des anti-invariants. Nous développons les systèmesde Hitchin sur ces espaces de modules et on les utilise pour dériver une classification de leurs composantes connexes en les dominant par des varietésde Prym. On étudieaussi le fibrédéterminant sur les espaces de modules des fibrésvectoriels anti-invariants. Dans certains cas, ce fibréen droites admet certaines racines carréesappeléesfibrés Pfaffiens. On montre que les espaces des sections globales des puissances de ces fibrésen droites (les espaces des fonctions theta généralisées)peuvent êtrecanoniquement identifier avec les blocs conformes associésaux algèbresde Kac-Moody affines twistéesde type A(2).

To my parents: Ammar & Zohra

vii Acknowledgements

”...and he [prophet Solomon] said: ”My Lord, enable me to be grateful for Your favor which You have bestowed upon me and upon my parents and to do righteousness of which You approve. And admit me by Your mercy into [the ranks of] Your righteous servants.””

Quran [27,19]

Undoubtedly, this work would not have been possible without the guidance and the support of my advisor Pr. Christian PAULY. He has introduced me to this beautiful domain of mathematics, and gave me a very interesting research topic. I would like to express my sincere appreciation and thanks for your continuous support and encouragement. Working under your supervision more than three years now was a very enjoyable and delightful experience. You have taught me a lot of things, even outside mathematics. Your advices on both research as well as on my career have been priceless.

My gratitude also goes to the members of the committee: Pr. A. BEAUVILLE, Pr. I. BISWAS, Pr. J. HEINLOTH, Pr. C. SIMPSON and Pr. C. SORGER. I would like to express my sincere thanks for acting as my committee members and for letting my defense be a very enjoyable moment. As well for your brilliant comments and suggestions.

From behind the scenes, there are two persons that I owe them an enormous debt of gratitude, Ammar & Zohra, my beloved parents. Thank you so much for your support and Do¨aa.Your favour is great, I hope I can give you back a small amount of what you have sacriﬁced for me since always.

My thanks goes also to Pr. Kamel BETTINA, thank you for your continuous support and encouragements, and for your priceless advices.

I am not going to forget my brothers and sisters, specially my twin Hossein. Thanks to all of you for everything. Also my thanks goes to my friends and colleagues specially Ana Pe´on-Nieto.

I would like to thank the administrative team of LJAD for their hospitality and kind- ness, specially Rosalba BERTINO, Julia BLONDEL, Isabelle DE ANGELIS and Clara SALAUN.

Contents

Abstract iii

Acknowledgements vii

Introduction1

1 Invariant Vector Bundles7 1.1 Invariant line bundles...... 7 1.2 Invariant vector bundles...... 9 1.3 Inﬁnitesimal study...... 11 1.4 Moduli space of σ−invariant vector bundles...... 13

2 Anti-invariant Vector Bundles 19 2.1 Anti-invariant vector bundles...... 19 2.2 Bruhat-Tits parahoric G−torsors...... 21 2.3 The existence problem...... 23 2.4 Moduli space of anti-invariant vector bundles...... 25 2.4.1 σ−quadratic modules...... 25 2.4.2 Semistability of anti-invariant bundles...... 28 2.4.3 Construction of the moduli space...... 31 σ−symmetric case...... 31 σ−alternating case...... 33 2.5 Tangent space and dimensions...... 35

3 Hitchin systems 41 3.1 Generalities on spectral curves and Hitchin systems...... 42 3.2 The Hitchin system for anti-invariant vector bundles...... 45 3.2.1 σ−symmetric case...... 50 The ramified case...... 50 The étalecase...... 53 Trivial determinant case...... 55 3.2.2 σ−alternating case...... 57 The ramified case...... 57 The étalecase...... 62 Trivial determinant case...... 63 3.3 The Hitchin system for invariant vector bundles...... 65 3.3.1 Smooth case...... 67 3.3.2 General case...... 67

4 Conformal Blocks 73 4.1 Preliminaries on twisted Kac-Moody algebras...... 73 4.2 Loop groups and uniformization theorem...... 77 4.2.1 Uniformization theorem...... 77 x

4.2.2 The Grassmannian viewpoint...... 78 4.2.3 Central extension...... 81 4.3 Determinant and Pfaﬃan line bundles...... 82 4.4 Generalized theta functions and conformal blocks...... 84 4.5 Application: An analogue of a result of Beauville-Narasimhan-Ramanan.. 87

A Anti-invariant vector bundles via representations 91

B Stability of the pullback of stable vector bundles and application 95

C On the codimension of non very stable rank 2 vector bundles 99

D Some Results on anti-invariant vector bundles 101 D.1 Anti-invariance of elementary transformations...... 101 D.2 Another description of anti-invariant vector bundles...... 102 D.3 Equality between two canonical maps...... 103

E Rank 2 case 105

F Lefschetz Fixed Point Formula 107 1

Introduction

Moduli spaces are one of the fundamental constructions of algebraic geometry. They arise in connection with classification problems. Roughly speaking a moduli space for a collection of objects A and an equivalence relation ∼ between these objects is a classification space such that each point corresponds to one, and only one, equivalence class of objects. Therefore, set theoretically, the moduli space is defined as the set of equivalence classes of objects A/ ∼. The study of moduli problems is a central topic in algebraic geometry. After the development of GIT theory, moduli spaces of vector bundles over curves were constructed in the 70’s by Mumford, Narasimhan and Seshadri. Since then, these moduli spaces have been intensively studied by many mathematicians.

Moduli problems of line bundles over complex curves have been studied in the 19th century by Weierstrass, Riemann, Abel, Jacobi and others. The Jacobian of a curve builds a bridge between the geometry of curves and the theory of abelian varieties. The analogue of the Jacobian for a cover of two curves, called the Prym variety, have attracted the attention of many mathematicians since Mumford’s seminal article in the 70’s. Prym varieties are deﬁned as the identity components of the kernel of the norm map attached to some cover X → Y . In the case of degree 2 covers, they have a special description as the identity component of the locus of isomorphism classes of line bundles L over X such that

σ∗L =∼ L−1, where σ is the involution on X that interchanges the two sheets.

In this dissertation, we study higher rank vector bundles with such an anti-invariance propriety. Consider the moduli space UX (r, 0) of stable vector bundles of rank r and degree 0 over a smooth projective curve X with an involution σ. This involution induces by pullback an involution on UX (r, 0). Let E be a stable vector bundle, we say that E is anti-invariant if there exists an isomorphism

ψ : σ∗E −→∼ E∗, where E∗ is the dual vector bundle. We say that the anti-invariant vector bundle E is σ−symmetric (resp. σ−alternating) if σ∗ψ = tψ (resp. σ∗ψ = − tψ). We denote σ,+ σ,− by UX (r) and UX (r) the loci of σ−symmetric and σ−alternating anti-invariant vector bundles respectively. We will see that these varieties correspond to moduli spaces of the form MY (G), i.e. moduli spaces of G−torsors over Y for some particular type of group schemes G called parahoric Bruhat-Tits group schemes (see for example [PR08a], [Hei10] and [BS14]). Parahoric group schemes are special case of integral models of a semisimple algebraic group G. An integral model of G over X is a smooth affine group scheme whose generic fiber is isomorphic to G. They have been introduced by Bruhat and Tits in their seminal work [BT72], [BT84]. For an integral model G, there is a finite number of points, called ramification points, over which the fiber of G is not semisimple. An integral model G is 2 Contents

parahoric if, for a ramification point p, the fiber GOp is a parahoric subgroup of GK ([BT84], Définition5.2.6), where Op is the completion of the local ring at p and K = Frac(Op). The name ”parahoric” is a portmanteau of ”parabolic” and ”Iwahori”. Roughly speaking, parahoric subgroups are the natural generalizations of the parabolic subgroups to groups defined over local fields. For example, the parabolic subgroups P of G are those such that G/P is proper. Similarly, a parahoric subgroup can be defined as a subgroup P of G(K ) such that G(K )/P is ind-proper (i.e direct limit of proper varieties). They can also be defined as the stabilizer of some self-dual periodic lattice chain (In some cases one should take the intersection with the kernel of the Kottwitz homomorphism, cf. [PR08b] §4).

The main problem considered in this thesis is the study of the moduli spaces of anti- invariant vector bundles. By studying the deformations of anti-invariant vector bundles, σ,± we identify the fibers of the tangent bundles to UX (r) with eigenspaces associated to the eigenvalues ∓1 with respect to the canonical involution on H1(X, End(E)). This involution is induced by the anti-invariance structure of E. Based on that, we use Lefschetz fixed point theorem to drive formulas for their dimensions. Namely we prove that 1 rn dim(U σ,±(r)) = (g − 1) ± , X 2 X 2 where gX is the genus of X and 2n is the degree of the ramification divisor of X → X/σ. After that, we introduce the definitions of semistability and S−equivalence of anti- invariant vector bundles (which is closely related to the semistability of orthogonal and symplectic bundles). Using a twisted notion of quadratic modules ([Sor93]), which we σ,+ call σ−quadratic modules, we construct the moduli space MX (r) that parameterizes the S−equivalence classes of semistable σ−symmetric anti-invariant vector bundles of rank σ,− r over X. The same method can be used to construct the moduli space MX (r) of semistable σ−alternating vector bundles, using the σ−alternating modules rather than the σ−quadratic ones.

Next, we consider the irreducibility problem. To study the connected components of σ,± UX (r), we use the Hitchin system and the theory of the nilpotent cone to establish dominant maps on these moduli spaces from some Prym varieties (we obtain similar results as in [BNR89]).

Hitchin systems are algebraically integrable systems defined on the cotangent space of the moduli space of stable G−bundles on a Riemann surface. They lie at the crossroads between algebraic geometry, Lie theory and the theory of integrable systems. They have been introduced and studied by Hitchin ([Hit87]) in the case of classical algebraic groups (GLr, Sp2m and SOr). Let MX (G) be the moduli space of stable G−bundles over X, the tangent space to MX (G) at a point [E] can be identified with 1 ∼ 0 ∗ H (X, Ad(E)) = H (X, Ad(E) ⊗ KX ) , where Ad(E) is the adjoint bundle associated to E, which is a bundle of Lie algebras isomorphic to g = Lie(G). Hence, by Serre duality, the fiber of the cotangent bundle at E 0 is H (X, Ad(E) ⊗ KX ). If we consider a basis of the invariant polynomials on g under the adjoint action, we get a map

k ∗ 0 M 0 di TEMX (G) = H (X, Ad(E) ⊗ KX ) −→ H (X,KX ), i=1 Contents 3

where the {di} are the degrees of these invariant polynomials. Hitchin ([Hit87]) has shown that these two spaces have the same dimension. In the case G = GLr, a basis of the invariant polynomials is given by the coeﬃcients of the characteristic polynomial. If E is a stable vector bundle, then this gives rise to a map

r 0 M 0 i HE : H (X, End(E) ⊗ KX ) −→ H (X,KX ) =: W, i=1 which associates to each Higgs ﬁeld φ : E → E ⊗ KX , the coeﬃcients of its characteristic polynomial. The associated map

∗ H : T MX (GLr) −→ W

2 is called the Hitchin morphism. By choosing a basis of W , H is represented by d = r (gX − 1) + 1 functions f1, . . . , fd. Hitchin has proved that this system is algebraically completely integrable, i.e. its generic fiber is an open subset of an abelian variety of dimension d, f1, . . . , fd are Poisson-commute, f1 ∧ · · · ∧ fd is generically nonzero and the vector fields ∗ Xf1 ,..., Xfd associated to f1, ··· , fd (defined using the canonical 2−form on T MX (GLr)) are linear. Moreover, consider the map

∗ Π: T UX (r, 0) → UX (r, 0) × W whose ﬁrst factor is the canonical projection and the second factor is H . Then, it is proved in [BNR89] that Π is dominant.

σ,+ We start by describing the basis of the Hitchin morphism on the spaces UX (r) and σ,− σ,+ σ,− σ,± UX (r), i.e. we deﬁne two subspaces W and W of W such that dim(W ) = σ,± dim(UX (r)) and the map Π induces, by restriction, maps

∗ σ,± σ,± σ,± Π: T UX −→ UX × W . Using the nilpotent cone theory, we show that these maps are still dominant. The space W σ,+ is actually a vector subspace of W . However, in the ramified case, W σ,− is not a vector subspace. It is in fact an affine subvariety given by some quadratic equations. In the unramified case, the two spaces coincide. Moreover, we study the smoothness of the spectral curves (see section 3.1) when the spectral data are in W σ,±; for general point in W σ,+, the associated spectral curve is smooth, and for general point in W σ,−, the associated spectral curve is singular with just nodes as singularity. In both cases, the involution σ lifts to an involution on the spectral curve. Based on a result of Beauville, Narasimhan and Ramanan ([BNR89]) we show that the Prym varieties on these general spectral curves (or their normalizations in the singular case), with respect to these lifting of σ, dominate our moduli spaces of anti-invariant vector bundles. Using these results, we deduce a complete classification of the connected components of σ,+ σ,− the loci UX (r) and UX (r). We also consider the case of trivial determinant anti-invariant vector bundles, denoted σ,± SU X (r). This case turns out to be a slightly different. For example, we will show that σ,− SU X (r) has a big number of connected components in the ramified case.

To sum up, by studying the Hitchin system on the moduli spaces of anti-invariant vector bundles, we deduce the following classiﬁcation of the connected components : 4 Contents

• If π is ramiﬁed, then

σ,+ σ,+ – UX (r) and SU X (r) are connected. σ,− – UX (r) has two connected components, when r is even (and empty otherwise). σ,− 2n−1 – SU X (r) has 2 connected components (when r is even), where 2n is the number of ﬁxed points of σ.

• If π is unramiﬁed, then σ,+ ∼ σ,− – UX (r) = UX (r) and each one has two connected components. σ,+ – SU X (r) is connected. σ,− – SU X (r) is connected if r is even, and empty otherwise.

We consider also the Hitchin system on the moduli spaces of stable σ−invariant vector bundles. A vector bundle E is called σ−invariant if σ∗E =∼ E. The moduli space of these vector bundles has a lot of connected components (at least in the ramiﬁed case) which are parameterized by some topological type naturally attached to the linearizations on the considered bundles at the ramiﬁcations points. By an elementary computation, we deduce the number of all these types. The σ−invariant vector bundles are a special case of the (π, G)−bundles, as called by Se- shadri ([Ses70], [BS14]). In our case of vector bundles (i.e. G = GLr) they correspond (in some sense) to parabolic vector bundles on the quotient curve Y := X/σ, where the parabolic structure is over the branch locus of the double cover X → Y , and this structure is encoded by the type of the σ−invariant bundles.

As in the anti-invariant case, we describe explicitly the base of the Hitchin map for any type of σ−invariant vector bundles. Moreover, for each type, we show that the invariant locus of the Jacobian varieties of the general spectral curve (or its normalization) dominate the moduli space of the σ−invariant vector bundles. We should mention that the Hitchin systems for parabolic vector bundles have been already studied (cf. for example [LM] where the case of smooth spectral curves is considered). How- ever, we consider the general case where the spectral data deﬁne singular spectral curves. We show that by considering the normalizations of these singular spectral curves we still get dominance results as in the smooth case.

Very recently, Baraglia, Kamgarpour and Varma have studied the complete integrability of the Hitchin system over the moduli spaces of parahoric G−bundles, for a non-twisted parahoric group scheme G. This can be thought of as a generalization of the parabolic bundles case. As far as we know, the Hitchin system for the twisted parahoric G−torsors has not been considered before. Our study however treats the spacial case of twisted parahoric group schemes of type A.

The next problem that we consider is the study of line bundles over these moduli spaces. The question that arises naturally is whether the restriction of the determinant σ,± σ,− bundle to SU X is primitive. Since using the Hitchin system we showed that UX (r) is dominated by a Prym variety of some unramiﬁed double cover and since the restriction of the polarization of the Jacobian to this Prym variety has a square root, this let us σ,− conjecture that the restriction of the determinant bundle to SU X (r) has a square root too. We will show that this is true and in fact these square roots are parameterized by the Contents 5

σ−invariant theta characteristic over the curve X. We call these square roots Pfaffian of cohomology line bundles. σ,+ On the other hand, the restriction of the determinant bundle to SU X (r) is primitive in the ramified case. This can be seen also using the results obtained from the study of the Hitchin system. σ,+ σ,− In the étalecase, the two spaces UX (r) and UX (r) are isomorphic, as we have mentioned above, and the Pfaffian line bundle exists also in this case.

We also consider the spaces of generalized theta functions of the powers of the determi- σ,+ σ,− nant and Pfaffian line bundles on SU X (r) and SU X (r) respectively. Using the results of Kumar and Mathieu ([Kum87], [Mat88]), we show that these vector spaces can be canonically identified with the conformal blocks of the twisted affine Kac-Moody Lie algebras, called twisted conformal blocks. In particular, we prove a special case of a conjecture by Pappas and Rapoport ([PR08b] Conjecture 3.7)

The twisted conformal blocks have been defined by Frenkel and Szczeny [FS04] in the framework of vertex algebras. However, in our case, one can defined them in the usual way; roughly, giving a ramified cover X → Y of degree d = 2 or 3, a level l, a simple Lie algebra g and a set of dominant weights (λp)p∈Ram(X/Y ) (labeled by the ramification divisor) of an affine twisted Kac-Moody Lie algebra Lˆ(g, τ) associated to an automorphism τ of g of order d. Then the twisted conformal blocks associated to this data is defined as the dual of the space of coinvariant of the product of the irreducible integral representations of level l τ associated to λp, with respect to the algebra g(X r R) . See section 4.1 for more details.

Plan of the thesis.

In the first chapter, we will start by studying the σ−invariant vector bundles. As we have mentioned, this is a special case of the (Γ, G)−bundles where in our case Γ is just Z/2. This theory has been studied by C.S. Seshadri ([Ses70], [Ses10] and [BS14]), also J.E. Andersen and J. Grove ([AG06]) has studied the invariant vector bundles of rank 2 under the action of an automorphism of the curve. We start by classifying their connected components and count their dimensions. We also spell out their identification with the parahoric G−bundles over Y . We use a result by Balaji and Seshadri to count differently the dimensions of these connected components in the case of special linear group.

The second chapter will be reserved to the σ−anti-invariant vector bundles. We start by giving some basic fact and count the dimensions of these moduli spaces. Than we show how to identify such bundles with the parahoric G−torsors over the quotient curve. We use this identiﬁcation to deduce some results about the moduli stacks of the anti-invariant vector bundles by applying some results of Heinloth ([Hei10]). We also construct the associated moduli spaces by introducing the σ−quadratic and σ−alternating modules.

In the third chapter we study the Hitchin system over the moduli spaces of anti-invariant vector bundles as well as the invariant ones. We prove that these Hitchin systems are still algebraically integrable in some cases. We use these systems to classify the connected σ,± σ,± components of UX (r) and SU X (r). This was in fact our motivation to consider these 6 Contents algebraic systems. Based on the results of Laumon ([Lau88]), we show that in the anti- invariant case, the Prym varieties over a general spectral curves dominates our moduli spaces, and in the invariant case, the invariant locus in the Jacobian varieties dominates the moduli of σ−invariant bundles. This chapter corresponds to a preprint (arXiv:1612.06910) and it has been already submitted to a journal.

The last chapter will be devoted to the study of line bundles over the moduli spaces of anti-invariant vector bundles with trivial determinant and their global sections called generalized theta functions. We prove that the restriction of the determinant bundle to these moduli spaces admit square roots in some cases. We prove an identification of the generalized theta functions and the twisted conformal blocks associated to some twisted affine Kac-Moody Lie algebras (of type A(2) with the notation of [Kac90]). We also count the dimension of the space of generalized theta function of level 1 of the Pfaffian line bundle by establishing an analogue of a result of Beauville, Narasimhan and Ramanan ([BNR89]). 7

Chapter 1

Invariant Vector Bundles

The ground ﬁeld is always assumed to be C. We denote by X a smooth irreducible projective curve of genus gX > 2, together with a non trivial involution σ : X → X. We denote by π : X → X/σ =: Y the quotient map, gY the genus of Y and JX , JY their respective Jacobians.

1.1 Invariant line bundles

Let R ⊂ X be the ramiﬁcation divisor of π : X → Y . Since X is smooth, all of the ramiﬁcation points are simple and their number is 2n, for some non-negative integer n. Moreover by Hurwitz formula we have

gX = 2gY + n − 1.

n −1 ∗ Denote by ∆ ∈ Pic (Y ) the line bundle on Y such that π∗OX = OY ⊕ ∆ . If η ∈ ker(π ), then ∗ π∗π η = π∗OX 2 ⇒ det(π∗OX ) ⊗ η = det(π∗OX ), ∗ hence ker(π ) ⊂ JY [2], where for an abelian variety A, we denote by A[r] the r−torsion points of A. From [Mum74], we know that if π is unramified, then ker(π∗) = {0, ∆}, and ∗ in this case ∆ ∈ JY [2], and if π is ramified, then π is injective. ∗ Consider the endomorphism u = 1 − σ of JX , and let P0 = Im(u) = ker(2 − u)0.P0 is called the Prym variety of the cover π : X → Y . However, in this thesis, by a Prym variety of a cover of curves q : X¯ → Y¯ we mean (unless otherwise explicitly mentioned) the kernel of the norm map Nm : JX¯ −→ JY¯ attached to q, which may be non-connected (hence it is not an abelian variety). Recall that the norm map Nm is defined, at the level P P of Weil divisors, by associating to i nipi the divisor i niq(pi). The abelian variety P0 is connected of dimension

gX − gY = gY + n − 1.

Let e2 : JY [2] × JY [2] → {±1} the bilinear skew-symmetric form induced by the principle polarization. If π is unramiﬁed, we set

G = {η ∈ JY [2]| e2(η, ∆) = 1},

∗ ⊥ and G = JY [2] if not (i.e G = (Ker(π )) with respect to e2). Let H = {(L, π∗L−1)| L ∈ G}.

In fact, H is the kernel of the morphism

∗ π ⊗ i : JY × P0 −→ JX , 8 Chapter 1. Invariant Vector Bundles

where i is the inclusion P0 ,→ JX . Moreover, we have (see loc. cit.)

JX ' JY × P0/H. ∗ g −q g +q Let p = dim(P0), and q such that |ker(π )| = 2 Y , so we get |G| = 2 Y . ∗ ∼ If L is a σ−invariant line bundle, i.e. σ L −→ L, then we claim that L ∈ JY × P0[2]/H. ∗ Indeed, write L = π M ⊗ F for some (M,F ) ∈ JY × P0, since L is σ−invariant, then F ∗ −1 2 σ is σ−invariant too. But σ F = F , so F = OX . The converse is obvious. Hence, if JX denote the locus of σ−invariant line bundles, then σ JX ' JY × P0[2]/H. 2p ∗ 2q Note that card(P0[2]) = 2 and card(π G) = 2 , we conclude that σ ∗ 2(p−q) JX ' π JY × (Z/2Z) . σ 2(n−1) So the number of connected components of JX is 2 when n > 1 and 1 when n = 0. σ ∗ In particular, if π is unramiﬁed, we have JX = π JY . 2(n−1) σ We are going now to describe explicitly these 2 connected components of JX . First, we recall an important lemma (due to Kempf, see [DN89]). Lemma 1.1.1. (Kempf’s Lemma) Let E be a vector bundle on X, with a linearization ∼ φ : σ∗E −→ E, i.e. ϕ◦σ∗ϕ = id. Then (E, φ) descends to Y (i.e E =∼ π∗F for some vector bundle F on Y and φ is the canonical associated linearization) if and only if φ acts as the identity on the ﬁber Ep, for any p ∈ R. As a consequence of this Lemma, we have the following

Corollary 1.1.2. The canonical line bundle KX of X descends to Y . Proof. By diﬀerentiating the involution σ : X → X we get a linear isomorphism dσ : −1 ∗ −1 2 KX → σ KX . Since σ = id, we deduce dσ ◦ σ∗(dσ) = id. −1 Hence dσ is a linearization of KX . Moreover if t is a local parameter near a ramiﬁcation point p ∈ R, then σ(t) = −t, hence dσ = −1 over p. By Lemma 1.1.1 we deduce that KX descends to Y .

∗ −1 By Hurwitz formula we have OX (R) = KX ⊗ π KY , it follows that OX (R) descends to Y . Furthermore, using the relative duality (see e.g [Har77] Ex III.6.10), we deduce that ∗ ∗ OX (R) = π ∆, hence KX = π (KY ⊗ ∆).

Remark 1.1.3. Suppose that π is ramified. Let L be a line bundle on Y , then π∗L has a canonical linearization. We call it the positive linearization, (because it equals +id over each p ∈ R). Its opposite is called the negative linearization. Moreover, fixing a linearization φ on a line bundle M induces an involution on the spaces Hi(X,M) (for i = 0, 1) defined by associating to a local section s the section φ(σ∗s). In the case of π∗L, we have, with respect to the positive linearization, the following identifications 0 ∗ ∼ 0 0 ∗ ∼ 0 −1 H (X, π L)+ = H (Y,L),H (X, π L)− = H (Y,L ⊗ ∆ ), where, for a vector space V with an involution, we denote by V+ (resp. V−) the eigenspace associated to the eigenvalue +1 (resp. −1). ∗ ∗ If π : X → Y is étale,then KX = π KY = π (KY ⊗ ∆). We define the positive linearization on KX to be the linearization attached to KY ⊗ ∆. 1.2. Invariant vector bundles 9

σ We will describe explicitly the connected components of JX . Consider S ⊂ R a subset −s ∗ σ ∗ of cardinality 2s, then for all M ∈ Pic (Y ), one has π M(S) ∈ JX , and it lies in π JY if and only if S = ∅ or S = R (Kempf’s lemma). Moreover, π∗M(S) and π∗N(T ) belong to the same connected component if and only if their diﬀerence is in the identity component ∗ ∗ ∗ c π JY . In other words, π L(S −T ) ∈ π JY , (for some L in Pic(Y )), hence S = T or S = T , where T c = R − T . The number of such subset S up to complementary is given by

n 1 X 2n 1 = 22n−1 = 22(n−1). 2 2k 2 k=0

σ Therefore, the connected components of JX are classiﬁed by the even cardinality subsets of R up to complementary.

The case of degree 1 line bundles is almost the same, the σ−invariant locus is denoted 1 σ 1 σ by Pic (X) . If π is unramiﬁed, then Pic (X) = ∅, so we assume that n > 1. Let p ∈ X be a ramiﬁcation point, the translation map given by

σ 1 σ Tp : JX −→ Pic (X)

L −→ L(p) = L ⊗ O(p). is an isomorphism. In particular Pic1(X)σ contains the same number of connected com- σ ponents as JX . As before, let S ⊂ R be a subset of cardinality 2s + 1 and M ∈ Pic−s(Y ). It is clear that π∗M(S) ∈ Pic1(X)σ, and if π∗M(S) and π∗N(T ) are in the same connected component, ∗ ∗ c then π L(S − T ) ∈ π JY which implies, as we have seen, that S = T or T . To ﬁnish, it is easy to see that the number of such subset of odd cardinality up to complementary is again 22(n−1). We mention that these line bundles has been already studied by Beauville in [Bea13].

σ Another method to identify the connected components of JX is to observe that P0[2] = σ σ JX [2], and in fact P0[2] intersects all the connected components of JX . Hence

σ ∗ π0(JX ) = P0[2]/π JY [2].

1.2 Invariant vector bundles

A vector bundle E on X is called σ−invariant if there exists an isomorphism

ϕ : σ∗E −→∼ E.

∗ The isomorphism ϕ is called linearization if ϕ ◦ σ ϕ = idE. In fact, a linearization corresponds to a lifting of the involution σ to an involutionσ ˜ : E → E, such that the following diagram E σ˜ ϕ−1 ! σ∗E / E

# σ X / X 10 Chapter 1. Invariant Vector Bundles commutes. Using the linearization ϕ we obtain a linear involution on the space Hi(X,E) for i = 0, 1, given locally by s −→ ϕ(σ∗s). i We denote their eignespaces by H (X,E)±. Remark 1.2.1. If E is σ−invariant and stable, then it has only 2 linearizations; ϕ and −ϕ.

Suppose that E is a σ−invariant stable vector bundle and ϕ : σ∗E → E a linearization. We deﬁne the type of E to be

τ = (ϕp)p∈R mod ± Ir, with ϕp ∈ End(Ep). We denote usually by kp the multiplicity of the eigenvalue −1 of ϕp, and most of the time we identify the type τ with the associated vector (kp)p∈R. Note that the vectors (kp)p and (r − kp)p represent the same type (due to multiplication by −1). Moreover, we have the following relation between the type and the degree d of E X kp ≡ d mod 2. p∈R

Indeed, deﬁne F to be the kernel of M 0 → F → E → (Ep)− → 0. p∈R

F is called negative elementary transformation of E. By Kempf’s Lemma, it follows that F descends to Y , hence X d − kp = deg(F ) ≡ 0 mod 2. p∈R

One can also deduce this relation by looking at the determinant of E, which is σ−invariant.

σ,τ Denote by UX (r, d) ⊂ UX (r, d) the locus of classes [E] ∈ UX (r, 0) such that E is σ,τ σ−invariant stable vector bundle of type τ. Note that UX (r, d) is smooth. In fact there is a more general result

Lemma 1.2.2. Let Z be a smooth variety with an involution τ. Then the ﬁxed locus Zτ is smooth closed subvariety of Z.

In fact the action can be linearized locally around any point z ∈ Zτ . This is true in more general context (see Edixhoven [Edi92]). By an elementary calculation, we get the number of all possible types:

1 2n  (r + 1) − 1 + 1 if r ≡ d ≡ 0 mod 2 4   σ 1 2n π0(UX (r, d)) = (r + 1) − 1 if r ≡ d + 1 ≡ 0 mod 2 4  1  (r + 1)2n if r ≡ 1 mod 2. 4 To prove the existence of stable σ−invariant vector bundles of a given type, we use cyclic covers. 1.3. Inﬁnitesimal study 11

2n P Lemma 1.2.3. Let τ = (kp)p∈R ∈ N such that p∈R kp ≡ d mod 2. Then there exists σ a stable σ−invariant vector bundle (E, φ) ∈ UX (r, d) of type τ.

Proof. Let β ∈ JX [r] be a primitive r−torsion line bundle over X which descends to Y . Denote by q : Xβ → X the cyclic unramified cover of degree r of X defined by β (see section 3.1). By Lemma 2.3.1, the involution σ : X → X lifts to an involutionσ ˜ on Xβ. Moreover, the fixed locus of this involution equals q−1(R). In particular, there are 2rn fixed points byσ ˜. Let R = {p1, ··· , p2n} and for each i we choose an order on the fiber −1 q (pi) = {pi1, ··· , pir}. Now for a type τ = (kp)p∈R of σ−invariant vector bundles, we define a typeτ ˜ = (kpij )i,j ofσ ˜−invariant line bundles on Xβ as follows : ( −1 if 1 6 j 6 ki kp = ij +1 otherwise .

It is easy to see that the direct image of aσ ˜−invariant line bundle of typeτ ˜ is a σ−invariant vector bundle of type τ. Moreover, by Proposition 2.3.2, we deduce that for general σ˜−invariant line bundle on Xβ, q∗L is in fact stable.

1.3 Inﬁnitesimal study

2 Recall that a deformation of E aver Spec(C[ε]) (ε = 0) is deﬁned to be a locally free coherent sheaf E on Xε = X × Spec(C[ε]) together with a homomorphism E → E of

OXε −module such that the induced map E ⊗OX → E is an isomorphism. Canonically, the 1 ∼ set of deformation of E over Spec(C[ε]) is isomorphic to H (X, End(E)), where End(E) = E ⊗ E∗ stands for the sheaf of endomorphisms of E. By deﬁnition, a deformation is locally free, so it is ﬂat, thus taking the tensor product with E of the exact sequence ε 0 → OX → OXε → OX → 0 we obtain the exact sequence

ε 0 → E → E → E → 0.

Assume now that E is stable σ−invariant vector bundle of rank r and degree d. Let τ σ,τ be the type of E. We want to identify the tangent space to UX (r, d) at E. The tangent space to the moduli space UX (r, d) at a smooth point E is given by ∼ 1 TEUX (r, d) = H (X, End(E)).

The linearization ϕ : σ∗E → E induces a linear involution f on H1(X,E ⊗ E∗) deﬁned locally by associating to local section x ⊗ λ of E ⊗ E∗ the section

f(x ⊗ λ) = ϕ(σ∗(x)) ⊗ σ∗( tϕ(λ)).

Clearly, this involution does not depend on the choice of ϕ. 1 ∗ Given η = (ηij)ij ∈ H (X,E ⊗ E ), it corresponds to an inﬁnitesimal deformation

0 → E → E → E → 0 over Xε. In fact if we set −1 gij = φi ◦ (id + εηij) ◦ φj , r where φi : E|Ui → Ui × C are some local trivializations of E, then (gij)ij are transition functions of E (we will prove this in §2.5, Lemma 2.5.1 below). 12 Chapter 1. Invariant Vector Bundles

σ,τ Now η ∈ TEUX (r, d) if and only if E is σ−invariant, and E is σ−invariant if and only if it has σ−invariant transition functions. Since we can choose φi to be σ−invariant, i.e. ∗ σ φi = φi ◦ ϕ, we deduce that E is σ−invariant iﬀ η is invariant with respect to f. Thus σ,τ ∼ 1 ∗ TEUX (r, d) = H (X,E ⊗ E )+. σ,τ In particular we deduce the dimension of UX (r, d). Proposition 1.3.1. The dimension of the locus of σ−invariant vector bundles of ﬁxed type τ is given by

σ,τ 2 X dim(UX (r, d)) = r (gY − 1) + 1 + kp(r − kp), p∈R where (kp)p∈R are the integers associated to τ. 1 ∗ Proof. To calculate the dimension of H (X,E ⊗E )+ we use Lefschetz fixed point theorem (cf. AppendixF), to simplify the notations we let 1 1 ∗ h± = dimC H (X,E ⊗ E )± . We have  1 1 2 h+ + h− = r (gX − 1) + 1 (By Riemann-Roch Formula) 1 , h1 − h1 = 1 − P Tr(f ) (By Lefschetz fixed point theorem)  + − 2 p∈R p 0 ∗ we have used the fact that h (X,E ⊗ E )+ = 1 (the identity E → E is σ−invariant). By the very definition, fp = ϕp ⊗ ϕp, it follows that the multiplicity of the eigenvalue −1 2 of fp is 2kp(r − kp), hence T r(fp) = (r − 2kp) , so we have ( h1 + h1 = 2r2(g − 1) + r2n + 1 + − Y . 1 1 2 P h+ − h− = 1 − r n + 2 p∈R kp(r − kp) It follows σ,τ 1 2 X dim(UX (r, d)) = h+ = r (gY − 1) + 1 + kp(r − kp). p∈R

σ,τ σ,τ˜ kp In particular, since det : UX (r, 0) → Pic (X) is surjective, whereτ ˜ = ((−1) )p∈R mod ± 1, we have σ,τ σ,τ dim(SU X (r)) = dim(UX (r, 0)) − gY 2 X = (r − 1)(gY − 1) + kp(r − kp). p∈R σ Remark 1.3.2. The dimension of the locus of σ−invariant vector bundle UX (r, d) is the maximum of these dimensions : ( 2 r2(g − 1) + n r + 1 r ≡ 0 mod 2 dim(U σ (r, d)) = Y 2 . X 2 r2−1 r (gY − 1) + n 2 + 1 r ≡ 1 mod 2 These dimensions correspond to the following types (called maximal types) ( {τ = (ϕ ) mod ± I | k = r/2, ∀p ∈ R} r ≡ 0 mod 2 MAX = p p r p . {τ = (ϕp)p mod ± Ir | kp = (r + 1)/2 or kp = (r − 1)/2} r ≡ 1 mod 2 In the odd case, the cardinal of MAX is 22(n−1). 1.4. Moduli space of σ−invariant vector bundles 13

1.4 Moduli space of σ−invariant vector bundles

We start here by recalling some results from [BS14]. A σ−group scheme over X is a group scheme G over X with a lifting of the action of σ to G as group scheme automorphism. Denote by MX (G) the moduli stack of G−torsors over X. Definition 1.4.1 ((σ, G)−bundle). Let G be a σ−group scheme over X.A(σ, G)−bundle is a G−bundle E over X with a lifting of the action of σ : X → X to the total space of E (denoted also by σ) such that for each x ∈ E and g ∈ G, we have σ(x · g) = σ(x) · σ(g). By definition, the action of σ on E is not a G−morphism. But it gives an isomorphism of total spaces (by the universal property of the fiber product)

ϕ E ×X X / E ,

$ X which veriﬁes ϕ(v · g) = ϕ(v) · σ(g), for g ∈ G ×X X and v ∈ E ×X X. This is again not a G−morphism, but we can associate to it canonically a G-isomorphism (over the identity of X) E −→∼ Eσ,

σ G where E = (E ×X X) × G, here G acts on itself via σ. Definition 1.4.2. (Parahoric group scheme) A smooth affine group scheme G over a curve X is said to be parahoric Bruhat-Tits group scheme if there is a finite subset R ⊂ X such that if Ox is the completion of the local ring at x ∈ R then GOx is a parahoric group scheme over Spec(Ox) (in the sens of Bruhat-Tits, [BT84] Définition5.2.6) for each x ∈ R and the fibers Gy is semisimple for all y ∈ X r R. In the following lemma, we show the correspondence between σ−invariant vector bundles and (σ, G)−bundles. Lemma 1.4.3. Giving a σ−invariant vector bundle (E, φ) of type τ is the same as giving (σ, Hτ )-bundle, for some σ−group scheme Hτ over X attached to τ.

Proof. Fix a σ−linearized vector bundle (Fτ , φτ ) of type τ and let Hτ = Aut(Fτ ). The linearization φτ induces an action on Hτ given by ∗ −1 g −→ στ (g) = φτ ◦ σ g ◦ φτ , this makes Hτ a σ−group scheme over X. Now let (E, φ) be a σ−invariant vector bundle of type τ, then the frame bundle E˜ := Isom(Fτ ,E) is clearly a (στ , Hτ )−bundle, where the ˜ ∗ −1 action of σ is given as follows: for a local isomorphism f ∈ E|U , we have σ(f) = φ◦σ f◦φτ . Conversely, giving (σ, Hτ )−bundle E˜, we have a commutative diagram

σ˜ E˜ / E˜

σ X / X, which gives us an isomorphism ∼ E˜ ×X X −→ E.˜ r Thus E = E˜(C ) is a σ−invariant vector bundle. 14 Chapter 1. Invariant Vector Bundles

σ,τ Let UX (r, d) be the moduli stack of σ−invariant vector bundles over X of type τ. In σ,τ the paper [BS14], they identify UX (r, d) with the stack of Gτ −torsors over Y σ,τ ∼ UX (r, d) = MY (Gτ ) for some parahoric Bruhat-Tits group scheme Gτ associated to the type τ. σ More precisely, consider a σ−group scheme H over X. Let G = ResX/Y (H) be the invariant subgroup scheme of the Weil restriction of H with respect to π : X → Y , i.e. σ the scheme that represents the functor π∗(H) (this is representable because π∗(H) is, see [BLR90] Theorem 4 and Proposition 6).

Theorem 1.4.4. [BS14] Let MX (σ, H) be the moduli stack of (σ, H)−bundles over X, then we have an isomorphism

∼ MX (σ, H) −→ MY (G)

σ given by the invariant direct image π∗ .

To apply this in our situation, let Hτ be the σ−group scheme deﬁned in Lemma 1.4.3. Then the group scheme Gτ is the σ−invariant Weil restriction of Hτ

σ Gτ = ResX/Y (Hτ ) .

Moreover, in loc. cit. the associated coarse moduli space is constructed and the main result assures that it is irreducible normal projective variety.

Remark 1.4.5. Since we deal with GLr−bundles, the parahoric group scheme Gτ is of parabolic type ([Ses10]), which implies that the moduli of σ−invariant vector bundles of type τ is isomorphic to the moduli space of parabolic vector bundles with parabolic structures, related to τ, at the branch points of X → Y . Indeed giving a σ−invariant vector bundle E of rank r, degree d and type τ, is the same as giving a vector bundle F of rank P r on Y of degree ν = d − p∈R kp, with a vector subspace Gp of Fπ(p) of dimension kp for each p ∈ R. To obtain F from E one can take the negative elementary transformation of E along the eigenspaces (Ep)−. Conversely, E can be constructing from F as the positive elementary transformation along the subspaces Gp. One veriﬁes easily that the dimension Q σ,τ of UY (r, ν) × p∈R Gr(kp,Fπ(p)) is the same as UX (r, 0), where Gr(kp,Fπ(p)) is the Grass- mannian parameterizing kp dimensional subspaces of Fπ(p). However, for a general reductive group G, the situation is more subtle.

Let’s spell out the deﬁnition of the semistability of σ−invariant vector bundles and give some properties.

Deﬁnition 1.4.6. We say that a σ−invariant vector bundle (E, φ) of rank r and degree d is semi-stable (resp. stable) if for every σ−invariant sub-bundle F of E one has

µ(F ) 6 µ(E) (resp. µ(F ) < µ(E)), where µ(E) = deg(E)/rk(E) is the slope.

Lemma 1.4.7. A σ−invariant vector bundle (E, φ) is semi-stable if and only if the vector bundle E is semi-stable. 1.4. Moduli space of σ−invariant vector bundles 15

Proof. the ”if” part is obvious. For a subbundle F ⊂ E we denote by

s(E,F ) = deg(E)rk(F ) − deg(F )rk(E).

Remark that µ(F ) 6 µ(E) if and only if s(E,F ) > 0, for any non-zero subbundle F of E. Let F be any subbundle of a semi-stable σ−invariant vector bundle (E, φ), let P be the image of σ∗F ⊕ F → E, and N ⊂ E such that

0 → N → σ∗F ⊕ F → P → 0.

Claim. The two sub-bundles P and N are σ−invariant. ∗ ∗ It is clear that φ(σ P ) ⊂ P , hence φ|P : σ P → P is a linearization. For N, as N is the kernel of the map σ∗F ⊕ F → P , and this map is clearly σ−equivariant for the action of σ, so the action of σ on σ∗F ⊕ F induces an action of N, thus it is σ−invariant. Now we can calculate

s(E,F ) = deg(E)rk(F ) − deg(F )rk(E) 1 1 = deg(E)(rk(P ) + rk(N)) − (deg(P ) + deg(N))rk(E) 2 2 1 = (s(E,P ) + s(E,N)) 2 > 0.

Recall the deﬁnition of semi-stability of (σ, GLr)-bundle (see [BS14] for example).

Deﬁnition 1.4.8. A(σ, GLr)−bundle is semi-stable (resp. stable) if for any maximal parabolic subgroup P ⊂ GLr and every σ−invariant reduction of structure group s : X → E(GLr/P ) we have ∗ deg(s E(glr/p)) > 0 (resp. > 0) where glr and p denote the Lie algebras of GLr and P respectively. r Proposition 1.4.9. A (σ, GLr)−bundle is (semi-)stable if and only if E(C ) is (semi- )stable σ−invariant vector bundle. Proof. (Adapted from [HM04]) r Let E be a (σ, GLr)−bundle. Suppose that E(C ) is semi-stable and let P ⊂ GLr be a maximal parabolic subgroup, and s : X → E/P a σ−invariant reduction of the structure group. The parabolic subgroup P corresponds to a ﬂag

r {0} ⊂ V ⊂ C . Denote F = (s∗E)(V ). ∗ ∼ ∗ r Claim. (1) s E(g/p) = F ⊗ (E(C )/F ). (2) F is σ−invariant.

Proof of the claim. 1. c.f Proposition 1 of [HM04].

2. Since P stabilizes V , F is well deﬁned, and since s is σ−invariant, s∗E is a (σ, P )−bundle. r Thus, F is a σ−invariant vector subbundle of E(C ). 16 Chapter 1. Invariant Vector Bundles

r r Thus µ(F ) 6 µ(E(C )), which is equivalent to µ(F ) 6 µ(E(C )/F ). Using the ﬁrst point of the claim, we deduce ∗ deg(s E(g/p)) > 0. Hence E is semi-stable as principal bundle. Conversely, assume that E is a semi-stable (σ, GLr)−bundle. Let F be a σ−invariant vector r subbundle of E(C ). By completing the transition functions of F to transition functions of E, we see that F is of the form s∗E(V ) for some reduction s to some maximal parabolic subgroup P ⊂ GLr, as F is σ−invariant, s is σ−invariant too. We deduce that

∗ deg(s E(g/p)) > 0. r As before, this implies that µ(F ) 6 µ(E(C )/F ), hence µ(F ) 6 µ(E), which means that r E(C ) is a semi-stable σ−invariant vector bundle. For the stability, one just has to replace the inequalities by strict ones.

As an application, we consider here the case G = SLr and we apply the main theorem of ∼ r−1 [BS14] to compute the dimension of the associated moduli space. Denote by T = (Gm) its maximal torus and SUr its maximal compact subgroup (of Hermitian matrices). Denote by ∗ ∗ h , i : X (SLr) × Y (SLr) → Z ∗ the canonical bilinear form on the spaces of characters X (SLr) and of 1−parameter sub- ∗ groups Y (SLr). Fix a type τ = (kp)p∈R mod ± 1, such that the kp > 0 are all even (because the vector bundles have trivial determinant). We associate to each kp the matrix

Ap = diag(−1, ··· , −1, +1, ··· , +1), | {z } kp times

∗ and a 1−parameter subgroup θ˜p ∈ Y (T)

θ˜p = (1, ··· , 1, 0, ··· , 0). | {z } kp times Finally let 1 θ = θ˜ ∈ Y ∗(T) ⊗ . p 2 p Q

(see [BS14] Lemma 2.2.8). Thus we can represent θp by

(1/2, ··· , 1/2, 0, ··· , 0). | {z } kp times

The root system associated to the adjoint representation of T is given by

R = {λi,j : T → Gm | λi,j(X) = xi/xj, i 6= j}

r As an element of Z , λi,j is equal to (0, ··· , 1, ··· , −1, ··· , 0) or (0, ··· , −1, ··· , 1, ··· , 0) (depending on whether i < j or j < i). We choose the set of simple root to be

S = {λi,i+1 | i = 1, . . . , r − 1} . 1.4. Moduli space of σ−invariant vector bundles 17

So that the set of positive roots are

+ R = {λi,j ∈ R | i < j}.

Note that the maximal root is given by

λ1,r = λ1,2 ··· λr−1,r.

We will count the dimension of this moduli space by applying the main theorem of [BS14].

σ,τ Theorem 1.4.10. Let SU X (r) be the moduli space of σ−invariant vector bundles with trivial determinant and of ﬁxed type τ = (kp)p mod ± 1 as above. Then the dimension of σ,τ SU X (r) is given by 2 X (r − 1)(gY − 1) + kp(r − kp). p∈R

Proof. To apply the main theorem of [BS14], we need to calculate the numbers e(θp) deﬁned by

e(θp) = dimR(SUr) − |S| − #{λ ∈ R | hθp, λi = ±1 or 0}.

It is easy to see that for any λi,j, one has 1 hθ , λ i = ± or 0. p i,j 2 r In fact hθp, λi,ji is just the dot product in Q of the two vectors θp and λi,j. The number of λi,j such that hθp, λi,ji = 0 is

2 2 (r − kp) + kp − r.

It follows

2 2 2 e(θp) = r − 1 − (r − 1) − ((r − kp) + kp − r) 2 2 2 = r − r − (r − r − 2rkp + 2kp)

= 2kp(r − kp).

Finally we get

1 X dim(M (G )) = dim(SL )(g − 1) + e(θ ) Y τ r Y 2 p p∈R 2 X = (r − 1)(gY − 1) + kp(r − kp). p∈R

Chapter 2

Anti-invariant Vector Bundles

2.1 Anti-invariant vector bundles

Fix an integer r > 2. Let E be a vector bundle E over X of rank r. E is called σ−anti-invariant (or simply anti-invariant) vector bundle if there exists an isomorphism

ψ : σ∗E −→∼ E∗.

If E is stable, then this isomorphism is unique up to a scalar. Take an isomorphism ψ : σ∗E −→∼ E∗, by pulling back with σ and taking the dual we get t(σ∗ψ): σ∗E −→∼ E∗. t ∗ ∗ So, there exists a non-zero λ ∈ C, such that (σ ψ) = λψ. By applying again σ and taking the dual on this last equality, we deduce λ2 = 1, thus λ = ±1. Denote by ψ˜ the non-degenerated bilinear form canonically associated to ψ defined as the composition ψ⊗id ˜ ∗ ∗ Tr ψ : σ E ⊗ E −−−−→ E ⊗ E −→ OX , where Tr is the trace map. Sometimes we use this bilinear form instead of ψ. Definition 2.1.1. We say that (E, ψ) is σ−symmetric (resp. σ−alternating) if λ = 1 σ,± (resp. λ = −1). We denote by UX (r) ⊂ UX (r, 0) the locus of isomorphism classes of stable σ−symmetric (resp. σ−alternating) vector bundles E. The case of trivial determinant is σ,± denoted SU X (r). Observation. If π is ramified and r ≡ 1 mod 2, then ψ is necessarily σ−symmetric. Proof. Indeed, let p be a ramification point, then ψ : σ∗E → E∗ induces an isomorphism ∗ ψp : Ep → Ep which is symmetric or alternating. But since r is odd, ψp is necessarily symmetric.

Note also that in the special case of rank 2, the σ−anti-invariant vector bundles with trivial determinant are the same as the σ−invariant vector bundles. Remark 2.1.2. A stable σ−alternating vector bundle does not necessarily have a trivial σ,± determinant (not like the symplectic case). Moreover, the determinant map det : UX (r) → Prym(X/Y ) is surjective (see Proposition 2.3.3). Assume for the moment that π is ramified. Let (E, ψ) be a stable σ−alternating ∗ vector bundle, then over a ramification point p ∈ R, ψp : Ep → Ep is an antisymmetric isomorphism. If we assume that E has trivial determinant and ψ as well, then the Pfaffian pf(ψp) of ψp is equal to ±1. For such anti-invariant vector bundle, we associate a type

τ = (pf(ψp))p∈R mod ± 1. We will see in the next chapter that these types classiﬁes the connected components of the σ,− locus SU X (r) of stable σ−alternating vector bundles with trivial determinant. 20 Chapter 2. Anti-invariant Vector Bundles

Let (E, ψ) be a σ−symmetric (resp. σ−alternating) anti-invariant vector bundle, such that E is polystable vector bundle. It is easy to see that E can be decomposed as

a !  b  c ! M ⊕fi M ⊕gj M ∗ ∗ ⊕hk E = Fi ⊕  Gj  ⊕ (Hk ⊕ σ Hk )k i=1 j=1 k=1 with Fi, Gj and Hk stable vector bundles (mutually non isomorphic), such that

• Fi are σ−symmetric (resp. σ−alternating).

• Gj are σ−alternating (resp. σ−symmetric).

• Hk are not σ−anti-invariant.

In particular, one remarks that gj > 2 for all j. The isomorphism ψ can be decomposed as well in the form

a b c ψ = ⊕i=1αi ⊕j=1 βj ⊕k=1 γk

⊕fi where αi (resp. βj, γk) are σ−symmetric (resp. σ−alternating) isomorphism on Fi ⊕gj ∗ ∗ ⊕hk (resp. Gj ,(Hk ⊕ σ Hk )k ).

Let’s treat the case of line bundles. Consider a line bundle L such that σ∗L =∼ L−1. ∗ ∼ ∗ ∗ ∼ Because we have L ⊗ σ L = π Nm(L), it follows that π Nm(L) = OX , hence two cases may happen: ∗ 1. If π is ramiﬁed, then π is injective, so Nm(L) = OY . ∗ 2. If π is ´etale, then the kernel of π is {OY , ∆}, so either Nm(L) = OY or Nm(L) = ∆.

Lemma 2.1.3. If L is a line bundle such that Nm(L) = OX then L is σ−symmetric. Assume that π is ´etale,then if Nm(L) = ∆ then L is σ−alternating. Proof. The line bundle L ⊗ σ∗L has a canonical linearization given by transposition. And the line bundle π∗Nm(L) has the canonical linearization (which we have called positive in the ramiﬁed case). These two linearizations are the same via the isomorphism

L ⊗ σ∗L =∼ π∗Nm(L). ∗ ∼ −1 Assume that Nm(L) = OY , the isomorphism σ L = L is in fact a global section of L ⊗ σ∗L, which is unique up to scalar multiplication. Then by Remark 1.1.3, we have

0 ∗ 0 ∗ H (X,L ⊗ σ L)+ = H (X, π Nm(L))+ 0 = H (Y, Nm(L)) = C. This implies that L is σ−symmetric. If π is ´etaleand Nm(L) = ∆, then it is clear that L is anti-invariant, and again by Remark 1.1.3 we have

0 ∗ 0 ∗ H (X,L ⊗ σ L)− = H (X, π Nm(L))− 0 = H (Y, Nm(L) ⊗ ∆) = C. Hence L is σ−alternating.

Note that in the ´etalecase and odd rank, the determinant of a stable σ−alternating vector bundle belongs to Nm−1(∆). 2.2. Bruhat-Tits parahoric G−torsors 21

2.2 Bruhat-Tits parahoric G−torsors

σ Let G = ResX/Y (SLr) be the invariant subgroup scheme of the Weil restriction of SLr, where SLr is the constant group scheme X × SLr over X and the action of σ on SLr is given by σ(x, g) = (σ(x), tg−1).

Fix a σ−alternating vector bundle with trivial determinant (Fτ , ψτ ) of type τ. Deﬁne Pτ = Aut(Fτ ). It is a group scheme over X which is ´etalelocally isomorphic to SLr. The ∗ ∗ τ isomorphism ψτ : σ Fτ → Fτ induces an involution, denoted σ , on Pτ given by

t −1 ∗ t −1 t α −→ ψτ ◦ σ ( α ) ◦ ψτ .

τ So (σ , Pτ ) is a σ−group scheme over X. Finally deﬁne the group scheme

στ Hτ = ResX/Y (Pτ ) .

Proposition 2.2.1. The group schemes G and Hτ are smooth affine separated group schemes of finite type which are parahoric Bruhat-Tits group schemes. If r > 3, G and Hτ are not generically constant. The set of y ∈ Y such that Gy and (Hτ )y are not semi-simple is exactly the set of branch points of the double cover π : X → Y . Proof. For the first part, we refer to [BLR90] Section 7.6, Proposition 5. As well as [Edi92] Proposition 3.5. Moreover, by [PR08b] §4, taking I = {0}, we deduce that G(Op) is a parahoric subgroup of G(Kp), where here Op is the completion of the local ring at the branch point p ∈ Y , and Kp its fraction field. Further we will see (cf. subsection 4.2.2) that for every p ∈ B, the flag variety G(Kp)/G(Op) (resp. Hτ (Kp)/Hτ (Op)) is a direct limit of symplectic (resp. special orthogonal) Grassmannian which is proper, hence these flag varieties are ind-proper. So G(Op) (resp. Hτ (Op)) is parahoric subgroup of G(Kp) (resp. Hτ (Kp)). We can calculate the fibers of G explicitly. Let x ∈ X r R (recall that R is the divisor of ramification points). Denote by y its image in Y . By definition, we have

−1 σ σ Gy = SLr(π (y)) = (SLr × SLr) , where σ(g, h) = (th−1,t g−1). So

t −1 ∼ Gy = {(g, g ) | g ∈ SLr} = SLr.

−1 Now, take p ∈ B, π (p) is, scheme theoretically, a double point, let us see it as Spec(C[ε]), with ε2 = 0, this gives −1 σ σ Gp = SLr(π (p)) = SLr(C[ε]) , where the action of σ on C[ε] is given by ε → −ε. So Gp is the group of elements (g, h) such that g + εh = t(g − εh)−1 = (tg − ε th)−1 = tg−1 + ε tg−1 thtg−1, and det(g + εh) = 1. t −1 t t t In other words g = g , gh = ( gh) and g + εh has determinant 1. So g ∈ SOr(C), and h is an r × r matrix such that tgh is symmetric. The last condition is equivalent to

t t det(Ir + ε gh) = 1 + εTr( gh) = 1. 22 Chapter 2. Anti-invariant Vector Bundles

t 0 Hence Tr( gh) = 0. It follows that Gp is isomorphic to SOr(C) × Symr(C) with group low 0 given by (g, h)(k, l) = (gk, gl + hk), where Symr(C) is the additive group of symmetric traceless matrices. We have a non split exact sequence:

0 0 → Symr(C) → Gp → SOr(C) → 1.

Note that Gp is not semi-simple.

Assume now that r is even. With the exact same computation as above we get ∼ (Hτ )p = SLr for p ∈ Y not a branch point, and for a branch point p we have

0 0 → ASymr,p → (Hτ )p → Spr → 0, where t t ASymr,p = {h ∈ Mr|T r(h) = 0,Mph = hMp = − (Mph)}, t −1 where Mp = ( ψτ )p and Spr is the symplectic group over C. σ,+ σ,τ Let SU X (r) (resp. SU X (r)) be the stack deﬁned by associating to a C−algebra R the groupoid of (E, δ, ψ), where E is a σ−symmetric (resp. σ− alternating of type τ) vector bundle over XR = X × Spec(R), δ a trivialization of det(E) and a σ−symmetric (resp. σ−alternating of type τ) isomorphism ψ : σ∗E −→∼ E∗ which is compatible (in the obvious sens) with δ.

Proposition 2.2.2. Let MY (G) (resp. MY (Hτ )) be the stack of right G−torsors (resp. Hτ −torsors) on Y , then MY (G) (resp. MY (Hτ )) is a smooth algebraic stack, locally of σ,+ σ,τ finite type, which is isomorphic to SU X (r) (resp. SU X (r)). Proof. The first part of the theorem is proved in [Hei10] Proposition 1. ∼ ∼ By Theorem 1.4.4, MY (G) = MX (σ, SLr). So it is sufficient to show MX (σ, SLr) = σ,+ SU X (r). Let S be a C−algebra, and (E, δ, ψ) be an element of MX (σ, SLr)(S). Consider the automorphism of the frame bundle E˜ := Isom(O⊕r ,E) given by XS

ψ˜(f) = t(ψ ◦ σ∗f)−1, for f ∈ E˜ (we identify σ∗(O⊕r ) =∼ O⊕r using the canonical linearization). Since σ∗ψ = tψ, XS XS we get ψ˜ ◦ ψ˜(f) = f, thus ψ˜2 = id, so ψ˜ is a lifting of the action of σ to E˜, and any other lifting diﬀers by an involution of O⊕r . Moreover, for g ∈ SL (O ), we have XS r XS

ψ˜(f · g) = ψ˜(f) · σ(g),

t −1 where σ(g) = g . Thus E˜ is (σ, SLr)−bundle. Conversely, a G−bundle E over YS gives, by Theorem 1.4.4, a (σ, SLr)−bundle over ˜ XS denoted again by E. Let ψ be the action of σ on E. Then

r SL r E(C ) := E × r C is σ−anti-invariant vector bundle. Let U be a σ−invariant open subset of XS such r ⊕r that E(C )|U is trivial and ﬁx a σ−invariant trivialization ϕ : OU → E(C)|U . Deﬁne 2.3. The existence problem 23

t ˜ −1 ∗ −1 ∗ r r ∗ ψ|U = ψ(ϕ) ◦ σ ϕ , then ψ is a σ−symmetric isomorphism σ E(C )|U → E(C ) . ∗ r r ∗ Gluing such local isomorphisms, we get an isomorphism ψ : σ E(C ) → E(C ) . Hence we σ,+ get an element of SU X (r)(S).

Now, let (E, ψ) be a σ−alternating vector bundle over XS. Consider the bundle

E˜ = Isom(Fτ ,E).

It is an Hτ −bundle. Moreover, ψ induces an automorphism ψ˜ on E˜ given by

t −1 t ∗ −1 t ψ˜(f) = ψ ◦ (σ f) ◦ ψτ .

τ Clearly this is an involution which makes E˜ a (σ , Pτ )−bundle. τ Conversely, a (σ , Pτ )− bundle gives, with exactly the same method as before, a σ−alternating vector bundle.

Proposition 2.2.3. We have π1(Gη) = 1 and π1((Hτ )η) = 1. Proof. We treat just the case of G. The other one is similar. Since π : X → Y is generically unramified, Xη is two points (to see this, note that K(X) is quadratic extension of K(Y ), so K(Y ) = K(X), and there is two embeddings of K(Y ) ,→ K(X) inducing the canonical inclusion K(Y ) ⊂ K(Y ), using the Gal(K(X)/K(Y )), this gives the two points). So by definition Gη is the invariant part of the action ofσ ˜ on SLr(η) × SLr(η), hence it can be identified with SLr(η), thus π1(Gη) = 1.

Corollary 2.2.4. The stacks MY (G) and MY (Hτ ) are connected. Proof. This follows from Proposition 2.2.3 applied to [Hei10] Theorem 2.

We will give another proof of this result using the Hitchin system. More precisely, we construct dominant rational maps from some Prym varieties to the loci of stable anti- σ,+ σ,− invariant bundles SU X (r) and SU X (r).

2.3 The existence problem

Here we construct examples of stable anti-invariant vector bundles. Let β ∈ JX [r] a primitive r−torsion point of the Jacobian which descends to Y , so in particular we assume that the genus gY of Y is at least 1. Denote by q : Xβ −→ X the associated cyclic ´etale cover of X of degree r which can be deﬁned as a spectral curve associated to the spectral data (0, ··· , 0, 1) (see section 3.1). Denote by ι a generator of the Galois group Gal(Xβ/X).

Lemma 2.3.1. The involution σ : X → X lifts to an involution σ˜ : Xβ → Xβ. Moreover, if r is even, there are two such lifting of σ such that one of them has no ﬁxed points, we denote it by σ˜−.

r Proof. The curve Xβ is a spectral curve given by the equation x − 1 = 0 in the ruled −1 surface P(OX ⊕ β ). As in the proof of Proposition 3.2.1, the positive linearization on β gives an involutionσ ˜ on Xβ that lifts σ. If r is even, then the negative linearization gives also a lifting of σ. One remarks that q(Fix(˜σ)) ⊂ Fix(σ), hence if π : X → Y is étale, then Xβ → Xβ/σ˜ is étaletoo. However, if r is even, the negative linearization has no fixed point because 0 is not a root of xr − 1 = 0. 24 Chapter 2. Anti-invariant Vector Bundles

Proposition 2.3.2. The line bundles of degree 0 on Xβ such that q∗L is not stable are those with non-trivial stabilizer subgroup of hιi.

Proof. This is true for any Galois cover, it is proved in the (unpublished) paper of Beauville ”On the stability of the direct image of a generic vector bundle”. 0 Let L ∈ Pic (Xβ) such that q∗L is not stable. Let F,→ q∗L be a stable subbundle of degree 0, it follows ∗ ∗ ∗ r−1 ∗ q F,→ q q∗L = L ⊕ ι L ⊕ · · · ⊕ (ι ) L, ∗ L j ∗ hence q F is of the form j∈J (ι ) L for some J $ {0, ··· , r − 1}. In particular both F and q∗L are semi-stable. On the other hand, The adjunction formula gives a non-zero map q∗F → (ιk)∗L for any k. As q∗F is semi-stable of degree 0, this map is surjective. Hence L j ∗ k ∗ j∈J (ι ) L → (ι ) L is surjective for any k. It follows that there exists k ∈ {1, ··· , r − 1} such that (ιk)∗L =∼ L. So ιk is in the stabilizer of L. Conversely, let L such that (ιk)∗L =∼ L for some 0 < k < r. Then the vector bundle ι∗L ⊕ · · · ⊕ (ιk)∗L is ι−invariant, so it descends to a vector bundle, say F , on X. As deg(F ) = 0, by adjunction, we deduce that F,→ q∗L, hence q∗L is not stable. Now we can construct some stable anti-invariant vector bundles.

Proposition 2.3.3. 1. There exist stable σ−symmetric anti-invariant vector bundles. If r is even or π is ´etale,then there exist stable σ−alternating vector bundles.

2. The determinant maps

σ,+ + −1 det : UX (r) → P = Nm (OY ), ( Nm−1(O ) r ≡ 0 mod 2 det : U σ,−(r) → P − = Y , X Nm−1(∆) r ≡ 1 mod 2 and π ´etale are surjective.

Proof. 1. We denote Yβ = Xβ/σ˜ and Zβ = Xβ/σ˜− if r is even. By Proposition 2.3.2 −1 we deduce that a general element in Nm (OY ) has a stable direct image which Xβ /Yβ β 0 is σ−symmetric. Let ∆β (resp. ∆β) be the 2−torsion point attached to Xβ → Zβ −1 (resp. Xβ → Yβ), then a general element in Nm (∆β) has a stable direct image Xβ /Zβ which is σ−alternating. If r is odd and π : X → Y is ´etale,also a general element in Nm−1 (∆0 ) has a stable direct image which is again σ−alternating. Note that Xβ /Yβ β being σ−symmetric or σ−alternating here is due to Lemma 2.1.3.

2. If π is ramified, or π is étaleand r is odd, then the second point is clear due to taking the tensor product of a fixed anti-invariant vector bundle by elements of P ±. Assume that π is étaleand r is even, taking the tensor product by elements of P ± does not make the determinant surjective, so we need to prove the existence of stable vector bundles whose determinants are in both connected components of P ±. But ± ± + −1 one remarks that Nm : P → P is surjective, where P = Nm (OY ) and Xβ /X Xβ /Yβ β − −1 r(r−1)/2 P = Nm (∆β). Since we have det(q∗L) = Nm (L) ⊗ β , we deduce Xβ /Yβ Xβ /X that the image of the determinant map intersects the two connected components of P + = P −. Taking now the tensor product with elements of the identity component of P + gives the result. 2.4. Moduli space of anti-invariant vector bundles 25

2.4 Moduli space of anti-invariant vector bundles

2.4.1 σ−quadratic modules This subsection is devoted to the study of the moduli of σ−quadratic modules, which will be used later in the construction of the moduli space of σ−symmetric anti-invariant vector bundles. Our main reference here is [Sor93]. Let W be a ﬁnite dimension vector space with an involution σ, and H a vector space. A σ−quadratic form is a linear map q : H −→ H∗ ⊗ W such that for all x, y ∈ H

q(x)(y) = σ(q(y)(x)).

A σ−quadratic module with values in W is a pair (H, q) as above. A map between two σ−quadratic modules (H, q) and (H0, q0) is a linear map f : H → H0 such that

q = ( tf ⊗ id) ◦ q0 ◦ f.

For a vector subspace V ⊂ H, we deﬁne its orthogonal to be

V ⊥σ = {x ∈ H |q(x, y) = 0 ∀y ∈ V }.

A σ−isotropic (resp. totally σ−isotropic) subspace V of (H, q) is a vector subspace such that V ∩ V ⊥σ 6= 0 (resp. V ⊂ V ⊥σ ). We will mainly use the notion of totally σ−isotropic as we will see later on.

Deﬁnition 2.4.1. The σ−quadratic module (H, q) is called semi-stable (resp. stable) if for any non-zero totally σ−isotropic vector subspace V ⊂ H we have

⊥ dim(V ) + dim(V σ ) 6 dim(H) (resp. <). Remark that a semi-stable σ−quadratic module is necessarily injective. Denote by Γ(H,W )σ the vector space of σ−quadratic forms q : H → H∗ ⊗ W , and let σ σ σ P (H,W ) = PΓ(H,W ) . The group SL(H) acts linearly in a natural way on Γ(H,W ) by associating to q the ( tg−1 ⊗id)◦q◦g−1. This action induces clearly an action on P (H,W )σ.

Proposition 2.4.2. A σ−quadratic module (H, q) is semi-stable (resp. stable) if and only if the point [q] ∈ P (H,W )σ is semi-stable (resp. stable) with respect to the action of SL(H).

Proof. We use Hilbert-Mumford criterion ([Pot97] Theorem 6.5.5) and we use also their notation for the weight. Assume that q is semi-stable σ−quadratic form on H, let λ be a non trivial one parameter subgroup of SL(H). Consider the eigenvalue decomposition of

s M H = Hi, i=1

−m where the restriction of λ(t) to Hi equals t i id, we assume also that m1 < ··· < ms. Since λ(t) ∈ SL(H), we have s X midim(Hi) = 0. i=1

Note that since λ is not trivial, there exists k such that mk < 0 6 mk+1. Now q decomposes as q = (qij)ij, where qij : Hi −→ Hj. It follows that the Hilbert-Mumford weight of q is equal to µ(λ, q) = −min{mi + mj | ∀(i, j) such that qij 6= 0}. 26 Chapter 2. Anti-invariant Vector Bundles

k Suppose that µ(λ, q) < 0 and let V = ⊕i=1Hi. Then

M ⊥σ V ⊕ Hi ⊂ V , i∈I where I = {i > k + 1 | mj + mi 6 0 for all j 6 k}. In particular V is totally σ−isotropic. Let l = max(I), so we get

k l ⊥σ X X ml+1 dim(V ) + dim(V ) > ml+1 dim(Hi) + ml+1 dim(Hi) i=1 i=1 k l X X > − midim(Hi) + ml+1 dim(Hi) i=1 i=1 s l X X = midim(Hi) + ml+1 dim(Hi) i=k+1 i=1 s X > ml+1 dim(Hi) = ml+1dim(H), i=1 which contradicts the semistability of q, hence µ(λ, q) > 0. Conversely, assume that for any 1−parameter subgroup λ we have µ(λ, q) > 0. Let V ⊂ H be a totally σ−isotropic subspace with respect to q, and denote by H1 a complementary ⊥σ ⊥σ subspace of V in V , and by H2 a complementary subspace of V in H, so we have H = V ⊕ H1 ⊕ H2. Consider the integers

m1 = 2dim(H) − 2dim(V ) − dim(H1),

m2 = dim(H) − 2dim(V ) − dim(H1),

m3 = −2dim(V ) − dim(H1).

Then we have m3 < m2 < m1 and

m1dim(V ) + m2dim(H1) + m3dim(H2) = 0.

Let’s consider the 1−parameter subgroup λ of SL(H) associated to the decomposition H = V ⊕ H1 ⊕ H2 with characters given by the weights m1, m2 and m3 (respecting the order of the decomposition). It follows that λ acts on q by the matrix

 0 0 t−m1−m3   0 t−2m2 t−m2−m3  . t−m1−m3 t−m2−m3 t−2m3

By deﬁnition, we deduce that

µ(λ, q) = −min{−2m2, −m1 − m3} = 2m2, and by hypothesis we have µ(λ, q) > 0. Hence m2 > 0, which is exactly

⊥ dim(V ) + dim(V σ ) 6 dim(H). 2.4. Moduli space of anti-invariant vector bundles 27

Let (H, q) be a semi-stable and non-stable σ−quadratic module, there exists a minimal ⊥σ totally σ−isotropic subspace H1 of H such that dim(H1) + dim(H1 ) = dim(H). We ⊥σ repeat this procedure after replacing H by H1 /H1 with its reduced σ−quadratic form. So we construct a ﬁltration

0 ⊂ H1 ⊂ H2 ⊂ · · · ⊂ Hk ⊂ H, of totally σ−isotropic subspaces such that

⊥σ (i) Hi/Hi−1 ⊂ Hi−1/Hi−1 are minimal totally σ−isotropic such that

⊥σ ⊥σ dim(Hi/Hi−1) + dim(Hi/Hi−1) = dim(Hi−1/Hi−1).

⊥σ (ii) Hk /Hk is stable. We deﬁne the σ−quadratic graded module associated to (H, q) to be

k−1 ⊥σ M ⊥σ gr(H, q) = Hk /Hk (Hi/Hi−1) ⊕ (Hi/Hi−1) , i=1 with the induced form. The integer k is called the length of the graded σ−quadratic module. Two σ−quadratic modules are said S−equivalent if they have isomorphic graded modules. Proposition 2.4.3. Let Q(H,W )σ = P (H,W )σ,ss//SL(H) be the geometric quotient of the subspace of semi-stable points P (H,W )σ,ss by SL(H). Then a point of Q(H,W )σ represents an S−equivalence class of σ−quadratic modules. Proof. The proof is the same as that of [Sor93] Proposition 2.5. We prove it in two steps: 1. First we prove gr(H, q) is in the closure of the orbit of q by showing that there exists a 1−parameter subgroup λ of SL(H) such that gr(H, q) = limt→0 λ(t)·q. We prove this by induction on k. If k = 0, that’s (H, q) is stable, there is nothing to prove. Assume the result for k − 1. Let (H, q) be a semi-stable σ−quadratic module with a graded ⊥σ module of length k. Choose a minimal totally σ−isotropic subspace H1 ⊂ H1 ⊂ H. ⊥σ ⊥σ Let H2 and H3 be (any) complements of H1 in H1 and H1 in H respectively. Then we have the following decomposition of q

H1 H2 H3 ∗ H1 ⊗ W 0 0 α ! ∗ 0 , H2 ⊗ W 0 q β ∗ ∗ ∗ ∗ ∗ H3 ⊗ W σ α σ β γ

0 for some σ−quadratic module q on H2 and some maps α, β and γ (this last veriﬁes σ∗γ∗ = γ). Clearly the graded module associated to q0 is of length k − 1 and we can 0 apply the induction hypothesis to obtain a 1−parameter subgroup λ of SL(H2) such 0 0 0 that limt→0 λ (t) · q = gr(q ). Finally deﬁne λ to be the 1− parameter subgroup of SL(H) given by H1 H2 H3 H1 t 0 0 ! 0 t −→ H2 0 λ 0 . −1 H3 0 0 t

We see immediately that limt→0 λ(t) · q = gr(H, q). 28 Chapter 2. Anti-invariant Vector Bundles

2. We show here that the orbit of a σ−quadratic graded module (H, q) is closed. Again we use induction on the length k. If k = 0, then q is stable. For every 1−parameter subgroup of SL(H), let q0 = limt→0 λ(t)·q. Since q is stable, its orbit is proper. So by the valuative criterion of properness, we deduce that q0 is in the orbit of q. Assume now the result for k − 1, let λ be a 1−parameter subgroup and assume that the limit q0 = limt→0 λ(t) · q exists. Let H = H1 ⊕ H2 ⊕ H3 be a decomposition as above. So q can be written H1 H2 H3 ∗ H1 ⊗ W 0 0 α ! ∗ 0 . H2 ⊗ W 0 q β ∗ ∗ ∗ ∗ ∗ H3 ⊗ W σ α σ β γ

Denote Hi(t) = λ(t)(Hi), and αt = λ(t)·α. The subspace H1(t) is totally σ−isotropic ∗ with respect to qt = λ(t) · q and the module H1 → H3 ⊗ W is stable. We can assume ∗ ⊥σ that λ(t) (for all t ∈ C ) stabilizes H1 and H1 = H1 ⊕ H2. Hence we can write λ(t)−1 in the form

H1 H2 H3 H1 f(t) g(t) h(t) ! H2 0 u(t) v(t) . H3 0 0 w(t)

Moreover, without changing qt, we can assume that det(f(t)) = det(u(t)) = det(w(t)) = t 1. It follows that αt = f(t)αw(t). Since α is stable, and since αt has a limit by assumption, it follows, by properness, that f(t) and w(t) have limits f0 and w0. Moreover, By the induction hypothesis, we deduce that u(t) has a limit u0. Now we can explicitly calculate βt and γt in function of g(t), h(t) and v(t) (with the coeﬃ- cients of qt) and we deduce the existence of limits of g(t), h(t) and h(t). This ends the proof.

2.4.2 Semistability of anti-invariant bundles Let (E, ψ) be an anti-invariant vector bundle over X. We say that a subbundle F of E is σ−isotropic if the induced map ψ : σ∗F → F ∗ is identically zero.

Deﬁnition 2.4.4. Let (E, ψ) be an anti-invariant vector bundle over X. We say that it is semi-stable (resp. stable) if for every σ−isotropic sub-bundle F of E, one has

µ(F ) 6 0 (resp. µ(F ) < 0). Proposition 2.4.5. (E, ψ) is semi-stable if and only if E is semi-stable vector bundle.

Proof. We follow the same lines of the proof of [Ram81] 4.2, page 155. The ”if ” part is obvious. Conversely, take F to be any sub-bundle of E. Deﬁne F ⊥σ to be the kernel of the surjective morphism:

∼ ∗ ∗ ∗ ∗ E −→ σ E σ F .

Note that F ⊥σ have the same degree as F , and F is σ−isotropic if and only if F ⊂ F ⊥σ . Then, the sub-bundle N of E generated by F ∩ F ⊥σ is σ−isotropic. Indeed, we have 2.4. Moduli space of anti-invariant vector bundles 29

N ⊂ F , so F ⊥σ ⊂ N ⊥σ , interchanging F and F ⊥σ we get F ⊂ N ⊥σ , hence N ⊂ N ⊥σ . Let M be the image of F ⊕ F ⊥σ in E. We have M = N ⊥σ , to see this, note that N ⊥σ contains F and F ⊥σ , so it contains M, but this two bundles have the same rank. Moreover we have

0 → N → F ⊕ F ⊥σ → M → 0, which implies also 0 → M ⊥σ → F ⊕ F ⊥σ → N ⊥σ → 0, we deduce that they have the same degree too. Hence they are equal. Therefore, deg(N) = deg(F ), but deg(N) ≤ 0 because it is σ−isotropic and (E, ψ) is semi-stable by hypothesis, so E is semi-stable as a vector bundle.

Let E be a σ−symmetric anti-invariant vector bundle, the following lemma generalizes the isotropic ﬁltration of self-dual vector bundle.

Lemma 2.4.6. There exists a ﬁltration of E of the form

⊥σ ⊥σ ⊥σ 0 = F0 ⊂ F1 ⊂ · · · ⊂ Fk ⊆ Fk ⊂ Fk−1 ⊂ · · · ⊂ F0 = E, where Fi are degree 0 sub-bundles of E (which are of course σ−isotropic) such that Fi/Fi−1 is stable vector bundle of rank > 1 for i = 1, . . . , k. Proof. The proof is similar to that of Lemma 1.9 of [Hit05]. The proof is a constructive one, we consider the set of all σ−isotropic subbundles of E, which contains 0 and E. If E is stable anti-invariant vector bundle, then it has no σ−isotropic proper sub-bundle of degree 0, and the ﬁltration is 0 ⊂ 0⊥σ = E. Otherwise, let F1 be a σ−isotropic sub-bundle of E of degree 0 and smallest rank (it is a stable vector bundle, because otherwise, a proper sub-bundle of F1 of degree 0 would be a σ−isotropic sub-bundle of E, contradicting the minimality of rk(F1)). Now, we repeat this procedure on E/F1 instead of E. Lemma 2.4.7. Consider the above ﬁltration, then we have

∗ ⊥σ ⊥σ ∼ ∗ ∗ ⊥σ ∼ ⊥σ ∗ σ (Fi−1/Fi ) = (Fi/Fi−1) , σ (Fk /Fk) = (Fk /Fk) , for i = 1, . . . , k.

⊥σ Proof. For i = 1, this is just the deﬁnition of F1 . Let i > 1, and consider

⊥σ ⊥σ 0 ⊂ Fi−1 ⊂ Fi ⊂ Fi ⊂ Fi−1 ⊂ E.

We have a commutative diagram

⊥σ i ∗ ∗ 0 / Fi−1 / E / σ Fi−1 / 0 O p1 p2

% ∗ ∗ σ Fi O

∗ ? ∗ σ (Fi/Fi−1) .

Since the composition p2 ◦ p1 ◦ i is identically zero, it follows that p1 ◦ i factorizes through ∗ ∗ ⊥σ ∗ ∗ σ (Fi/Fi−1) . The resulting map Fi−1 → σ (Fi/Fi−1) is nonzero map because otherwise 30 Chapter 2. Anti-invariant Vector Bundles

⊥σ ⊥σ Fi−1 ⊂ Fi , thus Fi/Fi−1 = 0 which contradicts the deﬁnition of the above ﬁltration. Its ⊥σ kernel contains Fi , so we obtain a nonzero map

⊥σ ⊥σ ∗ ∗ Fi−1/Fi → σ (Fi/Fi−1) .

But this two bundles are stable of the same rank and degree, so the last map has to be an isomorphism. For i = k, we have a nonzero map

⊥σ ∗ ⊥σ ∗ Fk /Fk → σ (Fk /Fk) ,

⊥σ otherwise Fk = Fk . So the same argument as before gives the result. The above lemma proves that the bundle

k σ M ⊥σ ⊥σ ⊥σ gr (E) = Fi/Fi−1 ⊕ Fi−1/Fi ⊕ (Fk /Fk) i=1 is an anti-invariant vector bundle. Moreover, it is σ−symmetric (resp. σ−alternating) if E is σ−symmetric (resp. σ−alternating).

Deﬁnition 2.4.8. The vector bundle grσ(E) is called the σ−graded bundle associated to (E, ψ). Two σ−anti-invariant vector bundles E and F are said to be S−equivalent if their associated σ−graded bundles are isomorphic.

Example 2.4.9. We give an example of two non-isomorphic σ−symmetric anti-invariant vector bundles which are S−equivalent. Let M be an element of PrymX/Y , and φ : σ∗M −→∼ M ∗. The vector bundle M ⊕2 with the σ−symmetric isomorphism

0 φ ψ = φ 0

1 ∼ 1 is a σ−symmetric anti-invariant vector bundle. Now for η ∈ Ext (M,M)− = H (X, OX )−, where the involution on this vector space is given by pullback by σ. Consider the associated extension of M by M 0 → M → E → M → 0. 1 Note that in rank 2 taking the dual does not change the extension class in H (X, OX ) because of the formula E∗ =∼ E ⊗ det(E)−1. Since η is a −1 eigenvector, E is anti-invariant. Indeed, by pulling back by σ we get the extension 0 / σ∗M / σ∗E / σ∗M / 0

' ' ' 0 / M −1 / E ⊗ M −2 / M −1 / 0. But E ⊗ M −2 is isomorphic E∗. Moreover, if η 6= 0 then E is not isomorphic to M ⊕2 (see subsection 2.5 for more details about the deformations of anti-invariant vector bundles). However, clearly E and M ⊕2 are S−equivalent as σ−anti-invariant vector bundles. 2.4. Moduli space of anti-invariant vector bundles 31

2.4.3 Construction of the moduli space σ−symmetric case Fix an ample σ-linearized line bundle (O(1), η) of degree 1 over X (in the ´etalecase, there are no such bundle, so one has to take degree 2 instead of degree 1, but this doesn’t produce any diﬀerence). We follow the method of [Sor93] to construct this moduli space. Let ν be some big integer such that for any semi-stable coherent sheaf E over X of rank r and degree 0, we have H1(X,E(ν)) = 0 and E(ν) is generated by global sections. Let m m F = OX (−ν) where m = rν + r(1 − gX ). Denote H = C .

Consider the functor

Quotσ : (algebraic varieties) → (sets) which associates to a variety T the set of isomorphism classes of (E, q, φ), where E is ∗ ∗ coherent quotient sheaf q : p1F → E over X × T ﬂat over T , and φ is class, modulo C , of σ−symmetric isomorphism σ∗E =∼ E∗ (σ acts only on X), such that, for each t ∈ T , Et is a semi-stable, σ−symmetric and locally free of rank r and q induces an isomorphism 0 H → H (X,Et(ν)). Two triplets (E, q, φ) and (F, p, ψ) are isomorphic if there exists an isomorphism f : E → F such that p = f ◦ q and ψ ◦ σ∗f = tf −1 ◦ φ (for some φ ∈ φ and ψ ∈ ψ). σ Let [E, q, φ] ∈ Quot (C), consider the diagram

∗ σ q ∗ H ⊗ OX / σ E(ν) / 0

φ t ∗ q ∗ 0 / E (ν) / H ⊗ OX (2ν) .

The composition h = tq ◦ φ ◦ σ∗q gives, at the level of global sections, a σ−quadratic form ∗ 0 H → H ⊗ W , where W = H (X, OX (2ν)) with an involution induced by the linearization on O(1). Hence we get a point h ∈ P (H,W )σ. This actually deﬁnes a transformation H : Quotσ −→ P (H,W )σ, where P (H,W )σ is seen as a functor by associating to a variety σ 0 ⊕m 0 T the space P (HT ,WT ) , where HT = H (X×T, OX×T ) and WT = H (X×T, OX×T (2ν)).

Proposition 2.4.10. Let (E, ψ) be a σ−symmetric vector bundle, and h its corresponding point of Γ(H,W )σ, then the following are equivalent: (a) The bundle E is semi-stable.

(b) h is semi-stable with respect to the action of SL(H). Moreover, (E, ψ) is stable if and only if h is stable. Proof. Assume that (E, ψ) is semi-stable, let V ⊂ H be a totally σ−isotropic. Denote by F and F 0 the subsheaves of E generated by V and V ⊥σ respectively. By Proposition 2.4.5 the induced vector bundle is semi-stable, hence by [Pot97] Proposition 7.1.1, for all subsheaf F of E, one has h0(F (m)) h0(E(m)) , rk(F ) 6 rk(E) 0 for m > ν large enough. By applying this to F and F , and then summing up, we deduce 0 0 0 0 h (F (ν)) + h (F (ν)) 6 h (E(m)), 32 Chapter 2. Anti-invariant Vector Bundles which is the same as ⊥ dim(V ) + dim(V σ ) 6 dim(H). Hence (H, h) is semi-stable. So by Proposition 2.4.2, h is semi-stable with respect to the action of SL(H). Conversely, suppose that h is semi-stable, then by Proposition 2.4.2,(H, h) is also semi- stable. Let F be a σ−isotropic subbundle of E, V = H0(F (ν)) and V 0 = H0(F ⊥σ (ν)). We have V 0 ⊂ V ⊥σ . Indeed, we have a commutative diagram

∼ V 0 / H0(E(ν)) / H0(σ∗E∗(ν)) / H∗ ⊗ W

' H0(σ∗F ∗(ν)) / V ∗ ⊗ W.

Since F is totally σ−isotropic, the composition V 0 → V ∗ ⊗ W is identically zero. Hence V 0 ⊂ V ⊥σ . Since we have also V ⊂ V 0, we deduce that V is totally σ−isotropic subspace of H. So we get

0 ⊥ dim(V ) + dim(V ) 6 dim(V ) + dim(V σ ) 6 dim(H). It follows

0 ⊥σ ⊥σ dim(V ) + dim(V ) = deg(F ) + rk(F )ν + deg(F ) + rk(F )ν + r(1 − gX )

= 2deg(F ) + rν + r(1 − gX ) 6 rν + r(1 − gX ) = dim(H).

Hence deg(F ) 6 0. This proves that E is semi-stable. 0 0 Now, let i > 0 and denote by Hi = H ⊗ H (OX (i)), Wi = H (OX (2ν + 2i)). For a σ−quadratic module (H, h), we denote by (Hi, hi) the σ−quadratic module obtained as follows: taking the tensor product with O(i) we obtain

H ⊗ O(i) −→ H∗ ⊗ O(i) ⊗ W −→ H∗ ⊗ O(i) ⊗ W ⊗ H0(O(i))∗ ⊗ H0(O(i)).

Than at the level of global sections we deduce

∗ 0 2 Hi −→ Hi ⊗ W ⊗ H (O(i)) −→ Hi ⊗ Wi, and the composition is denoted hi. Let Z ⊂ P (H,W )σ be the locus of σ−quadratic forms h such that

rk(hi) 6 r(ν + i − gX + 1), ∀ i > 0.

It is clear that Z contains the image of H(C). Moreover we have the following Theorem 2.4.11. Let Qσ ⊂ Z be the open of semi-stable points, then Qσ represents the functor Quotσ.

Proof. We need to prove that H induces an isomorphism of functor between Quotσ and σ the functor of points of Q . The main point is to show this for the C valued points. By σ Proposition 2.4.10, we deduce that the image of H(C) is contained in Q (C). Giving a σ point h ∈ Q , ﬁx a representative h of h. Taking the tensor product with OX (−ν) gives

h ∗ ev ∗ H ⊗ OX (−ν) −→ H ⊗ W ⊗ OX (−ν) −→ H ⊗ OX (ν). 2.4. Moduli space of anti-invariant vector bundles 33

Let F = Ker(ev ◦ h) and E = H ⊗ OX (−ν)/F . E doesn’t depend on the chosen representative of h and we have the following commutative diagram

0 / F / H ⊗ OX (−ν) / E / 0

ev◦h

∗ ∗ ∗ p ∗ ∗ 0 / σ E / H ⊗ OX (ν) / σ F / 0.

By deﬁnition, ev ◦ h vanishes over F , hence it factorizes through E giving an injective map ∗ ∗ t t t f : E → H ⊗OX (ν), since h is σ−symmetric, we deduce that p◦ev◦h = σ ( h◦ ev◦ p) = 0, so the map f gives a σ−symmetric morphism ψ : σ∗E → E∗, which is clearly injective. Let s be the rank of E and d its degree. By what we have just said we deduce d 6 0. From the condition deﬁning Z, we deduce that for all i

d + s(ν + i + 1 − gX ) 6 rk(qi) 6 r(ν + i + 1 − gX ),

∗ so in particular we deduce that r > s. But since q is semi-stable, the map q : H → H ⊗W is injective, hence H0(F (ν)) = 0. Thus the map H → H0(E(ν)) is injective and we deduce

r(ν + 1 − gX ) 6 d + s(ν + 1 − gX ), hence d > 0, thus d = 0. It follows that s > r, and so r = s. Hence ψ is surjective, thus (E, ψ) is a σ−symmetric vector bundles. Using the universal family over Qσ, one can make the above construction functorial which gives an inverse to H.

Consider the functor

σ,+ BunX (r) : (algebraic varieties) −→ (sets), that associates to a variety T the set of isomorphism classes of families (E , ψ) of rank r σ−symmetric anti-invariant vector bundles over X parameterized by T , such that Et is semi-stable for all t ∈ T .

σ,+ σ σ,+ Theorem 2.4.12. Consider the good quotient MX (r) = Quot (C)//SL(H). Then MX (r) σ,+ is a coarse moduli space for the functor BunX (r), which is a projective variety, and its underlying set consists of S-equivalence classes of semi-stable σ−symmetric anti-invariant vector bundles.

Proof. Consider a family (E , ψ) of σ−symmetric semi-stable bundles parameterized by a ∗ ∗ variety T , then for ν big enough, p2∗E (ν) and p2∗(σ E (ν)) are locally free, so by choosing local trivializations, we deduce a unique, up to an action of SL(H), map to Qσ . Thus we σ,+ get a morphism T −→ MX (r). This is obviously functorial in T . σ,+ σ A point a ∈ MX (r), corresponds by H to a point of Q //SL(H), this transformation respects the graded gr. Hence, using Proposition 2.4.3, we deduce that a represents an S−equivalence class of semi-stable σ−symmetric vector bundles.

σ−alternating case σ,− The construction of the moduli space MX (r) of semi-stable σ−alternating vector bundles follows the same method as the σ−symmetric case, using σ−alternating modules rather than quadratic ones. A module q : H → H∗ ⊗ W is σ−alternating if

q(x)(y) = −σ(q(y)(x)). 34 Chapter 2. Anti-invariant Vector Bundles

Similar results about semistability, ﬁltrations and S−equivalence of σ−alternating forms can be checked in this case too. We omit the details.

By Proposition 2.4.5 we have canonical forgetful maps

σ,+ MX (r) → U X (r, 0),

σ,− MX (r) → U X (r, 0), where U X (r, 0) is the moduli space of semi-stable vector bundles of rank r and degree 0 σ,± over X. The images of these maps are obviously U X (r). A natural question arises: what are the degrees of these maps? ∗ ∗ Remark 2.4.13. Note that the involution E → σ E is well deﬁned on U X (r, 0), since we have gr(σ∗E∗) = σ∗(gr(E))∗.

σ,+ σ,+ σ,− σ,− Proposition 2.4.14. The forgetful maps MX (r) −→ U X (r) and MX (r) −→ U X (r) are injective. In particular they are bijective.

Proof. We treat the σ−symmetric case. Let (E, ψ) be a σ−symmetric vector bundle, ∗ ∗ ∗ suppose that E is stable, so AutGLr (E) = C and ψ : σ E → E is unique up to scalar multiplication. The action of AutGLr (E) on these σ−symmetric forms is given by

f · ψ = ( tf)ψ (σ∗f).

∗ 2 If f = ξIdE, with ξ ∈ C , then this action is simply given by ψ → ξ ψ. It follows that this action is transitive, hence (E, ψ) and (E, λψ) are isomorphic as σ−symmetric vector bundles. If E is strictly semi-stable, using the decomposition of such polystable anti-invariant vector bundles given at the end of section 2.1, we can assume that E is of the form F ⊕d or (G⊕σ∗G∗)⊕d for stable anti-invariant vector bundle F and stable non-anti-invariant vector bundle G. Now, the set of σ−symmetric isomorphisms ψ : σ∗E → E∗ is equal to the locus of symmetric matrices of GLd(C) in both cases. Hence it is suﬃcient to use the fact that non-degenerated symmetric matrices can be decomposed in the form tM × M. This shows σ,+ that all the σ−symmetric isomorphisms on E deﬁne the same point in MX (r).

The case of vector bundles with trivial determinant is slightly diﬀerent. For simplicity we consider the forgetful maps just on the stable loci

σ,±,s σ,± SMX (r) −→ SU X (r).

σ,±,s Here SMX (r) is the locus of stable σ−symmetric or σ−alternating vector bundles in σ,± the moduli space SMX (r). Proposition 2.4.15. We have two cases:

σ,+ σ,+ (1) If r is odd, then the forgetful map SMX (r) −→ SU X (r) is injective. σ,± σ,± (2) If r is even, the forgetful map SMX (r) −→ SU X (r) is of degree 2. Proof. Let (E, ψ) be a σ−symmetric vector bundle with a trivialization of its determinant. ∗ Suppose that E is stable. As AutGLr (E) = C , we see that AutSLr (E) = µr, where µr th 2 is the group of r roots of unity. Remark that the map µr → µr, given by ξ 7→ ξ is a bijection if r is odd, and it is two-to-one on its image if r is even. 2.5. Tangent space and dimensions 35

(1) If r is odd, since E is stable, ψ : σ∗E → E∗ is unique up to scalar multiplication, as

det(ψ) = 1, the number of such isomorphisms is exactly r. The action of AutSLr (E) on these r σ−symmetric forms is given by

f · ψ = ( tf)ψ (σ∗f).

As f = ξIdE, for ξ ∈ µr, then we conclude as in the proof of Proposition 2.4.14 that this action is transitive.

(2) Assume r is even, with the same argument as above, we see that the action has two diﬀerent orbits. So E admits two non equivalent σ−symmetric forms. The same argument applies for the σ−alternating case.

Remark 2.4.16. Note that the above Proposition is similar to the situation of forgetful map of orthogonal bundles. See [Ser08].

2.5 Tangent space and dimensions

The tangent space to the moduli space UX (r, 0) at a smooth point E is canonically given by ∼ 1 TESU X (r) = H (X, End(E)), where End(E) =∼ E∗ ⊗ E stands for the sheaf of endomorphisms of E. σ,± We want to identify the tangent spaces to UX (r) at a point E. Before that recall that 2 a deformation of E over Spec(C[ε]) (ε = 0) is defined to be a locally free coherent sheaf E on Xε = X ×Spec(C[ε]) together with a homomorphism E → E of OXε −module, such that the induced map E ⊗ OX → E is an isomorphism. Canonically, the set of deformations 1 of E over Spec(C[ε]) is isomorphic to H (X, End(E)). As by definition, a deformation is locally free, so it is flat, thus taking the tensor product of the exact sequence

ε 0 → OX → OXε → OX → 0 with E we get ε 0 → E → E → E → 0. Let E be a σ−symmetric anti-invariant vector bundle and ψ : σ∗E =∼ E∗. Suppose that −1 ∼ E is given by the transition functions fij = ϕi ◦ ϕj : Uij → GLr, where the ϕi : EUi −→ r Ui × C are local trivializations of E. The covering {Ui}i of X is chosen to be σ−invariant, i.e. σ(Ui) = Ui (to get such covering, just pullback a covering of Y that trivializes both π∗OX and π∗E over Y ). Note that we can choose {ϕi} such that the diagram

∗ ∗ r σ ϕi : σ EUi / Ui × C

ψ σ×Ir tϕ−1 : E∗ U × r i Ui / i C commutes. Indeed, by taking an ´etale neighborhood U of each point x ∈ X, such that ˜ σ(U) = U, we can construct a frame (e1, ··· , er) of E|U on which the pairing ψ : E ⊗ ∗ σ E −→ OX is represented by the trivial matrix Ir. To construct such a frame, we apply the Gram-Schmidt process. As in this procedure, we need to calculate some square roots, that’s the reason why we have to work on the ´etaletopology. Moreover, we should mention that 36 Chapter 2. Anti-invariant Vector Bundles

∗ if we start with a frame (u1, ··· , ur) near x, it may happen that ψ˜(ui ⊗ σ ui)x = 0, in this ∗ case, we just replace ui with ui +uj, for some j > i such that ψ˜((ui +uj)⊗σ (ui +uj))x 6= 0. ∗ t −1 Taking such trivializations, we get transition functions fij such that σ fij = fij . We 1 ∗ know that the extension E (which corresponds to some η = {ηij}ij ∈ H (X,E ⊗ E )) is given by transition functions of the form

fij + εgij : Uij → GLr(C[ε]).

We want to ﬁnd the relation between these transition functions and η. First of all, in order that {fij + εgij}ij represents a 1−cocycle, we must have the two conditions ( g = 0 ii . gijfji + fijgjkfki + fikgki = 0

1 ∗ Now let η = {ηij}ij ∈ H (X,E ⊗ E ), which veriﬁes ηii = 0 and

ηij + ηjk + ηki = 0.

Each ηij can be seen as local morphism

ηij : E|Uij / E|Uij

ϕj ϕi & y r Uij × C .

−1 Denote by gij = ϕi ◦ ηij ◦ ϕj , we can rewrite the above conditions on η in the form

gii = 0,

−1 −1 −1 ϕi ◦ gij ◦ ϕj + ϕj ◦ gjk ◦ ϕk + ϕk ◦ gki ◦ ϕi = 0. −1 Composing by ϕi from the left and ϕi from the right, we get

−1 −1 −1 −1 gij ◦ ϕj ◦ ϕi + ϕi ◦ ϕj ◦ gjk ◦ ϕk ◦ ϕi + ϕi ◦ ϕk ◦ gki = 0

⇔ gijfji + fijgjkfki + fikgki = 0.

−1 Lemma 2.5.1. fij + εgij = ϕi ◦ (id + εηij) ◦ ϕj are transition functions of E .

1 ∗ Proof. Let η = {ηij}ij ∈ H (X,E ⊗ E ), locally the extension E is trivial, that’s ∼ E |Uiε = E|Ui ⊕ εE|Ui , x 7→ ($(x), x − si ◦ $(x)), where $ : E → E and si is a local section of $ on the local open set Ui, and Uiε =

Ui × Spec(C[ε]). This isomorphism is OXε −linear. Composing with the trivialization

∼ ϕi + εϕi : E|Ui ⊕ εE|Ui −→ OUi ⊕ εOUi , we get a trivialization ∼ φi : E |Uiε −→ OUiε , given by φi = ϕi ◦ $ + εϕi(id − si ◦ $). 2.5. Tangent space and dimensions 37

Remark that −1 −1 −1 φi = si ◦ ϕi − εsi(id − si ◦ $)si ◦ ϕi . So, we calculate the transition functions of E

−1 −1 −1 2 φi ◦ φj = (ϕi ◦ $ + εϕi(id − si ◦ $))(sj ◦ ϕj − εsj(id − sj ◦ $)sj ◦ ϕj ) (because ε = 0) −1 −1 = fij + ε(ϕi(id − si ◦ $)sj ◦ ϕj − ϕi ◦ $ ◦ sj(id − sj ◦ $)sj ◦ ϕj ) −1 = fij + ε(ϕi(sj − si)ϕj ) −1 = fij + ε(ϕiηijϕj )

= fij + εgij.

σ,+ Now η is in the tangent space to UX (r) at E if and only if the corresponding extension E is σ−symmetric anti-invariant vector bundle on Xε, where σ extended to an involution on Xε by taking σ(ε) = ε. On the transition functions, this means that

∗ t −1 σ (fij + εgij) = (fij + εgij) , which gives1 ∗ t −1 σ fij = fij , and

∗ t −1 t t −1 σ gij = − fij gij fij ∗ ∗ ∗ −1 t −1 t t ⇔ σ ϕi ◦ σ ηij ◦ σ ϕj = − ϕi ◦ ηij ◦ ϕj ∗ ∗ ∗ −1 ∗ −1 t ∗ −1 ⇔ σ ϕi ◦ σ ηij ◦ σ ϕj = −σ ϕi ◦ ψ ◦ ηij ◦ ψ ◦ σ ϕj ∗ −1 t ⇔ σ ηij = −ψ ◦ ηij ◦ ψ ∗ t −1 −1 t t t −1 ⇔ σ (ηij ◦ ψ ) = −ψ ◦ ηij = − (ηij ◦ ψ ).

Thus t −1 1 ∗ η ◦ ψ ∈ H (X,E ⊗ σ E)−. 1 ∗ where H (X,E ⊗ σ E)− is the proper subspace associated to the eigenvalue −1 of the involution of H1(X,E ⊗ σ∗E) given by

ξ → σ∗( tξ).

σ,− Consider the case of UX (r). Assume that r is even and π is ramiﬁed (we will show σ,+ ∼ σ,− σ,− ˜ later that UX (r) = UX (r) in the ´etalecase). Fix a point E of UX (r). In this case, ψ can be represented with respect to some frame by the matrix 0 Ir Jr = . −Ir 0

Such frame gives a set of trivializations {ϕi}i such that

∗ t −1 ∗ (σ × Jr) ◦ σ ϕi = ϕi ◦ σ ψ.

1 −1 −1 −1 −1 −1 Recall that (f + εg) = f − εf gf in GLr(C[ε]), and det(f + εg) = det(f)(1 + εT r(f g)). 38 Chapter 2. Anti-invariant Vector Bundles

So the associated transition functions {fij} verify

∗ t −1 σ fij = −Jr fij Jr.

σ,− It follows that the deformation E is in the tangent space TEUX (r) if and only if we have ∗ t −1 σ (fij + εgij) = −Jr (fij + εgij) Jr t −1 t −1 t t −1 = −Jr fij Jr + εJr fij gij fij Jr. Thus

∗ ∗ ∗ −1 t −1 t t σ ϕi ◦ σ ηij ◦ σ ϕj = Jr ϕi ◦ ηij ◦ ϕjJr ∗ −1 t ⇔ σ ηij = −ψ ◦ ηij ◦ ψ ∗ t −1 t t −1 ⇔ σ (ηij ◦ ψ ) = (ηij ◦ ψ ). Finally t −1 1 ∗ η ◦ ψ ∈ H (X,E ⊗ σ E)+. We have showed so far Theorem 2.5.2. With the above notations, we have

σ,+ 1 ∗ (a) The tangent space to UX (r) at a point E is isomorphic to H (X,E ⊗ σ E)−. In particular we have r2 nr dim(U σ,+(r)) = (g − 1) + X 2 X 2 r(r + 1) = r2(g − 1) + n . Y 2

σ,− 1 ∗ (b) The tangent space to UX (r) at a point E is isomorphic to H (X,E ⊗ σ E)+. In particular we have r2 nr dim(U σ,−(r)) = (g − 1) − X 2 X 2 r(r − 1) = r2(g − 1) + n . Y 2

Proof. We need just to calculate the dimensions. Let E be a σ−anti-invariant stable vector bundle, denote by F = E ⊗ σ∗E. First we have

1 1 1 2 h (X,F ) = h+ + h− = r (gX − 1) + 1, (2.1)

0 0 1 1 where we denote for simplicity h± = h (X,F )±, h± = h (X,F )±. Let ς : σ∗F → F be the canonical linearization which equals to the transposition (σ∗(s ⊗ σ∗t) −→ t ⊗ σ∗s). Applying Lefschetz ﬁxed point formula (see AppendixF, also [AB68]), we obtain

1 1 0 0 X Tr(ςp) h+ − h− = h+ − h− − . det(id − dpσ) p∈R

It is clear that dpσ : TpX → TpX is equal to −id (see Lemma 1.1.2), and the trace of the involution ςp : Fp → Fp is equal to

dim(Fp)+ − dim(Fp)−. 2.5. Tangent space and dimensions 39

2 V2 0 0 But, Fp = Ep ⊗ Ep = Sym Ep ⊕ Ep, and h+ = 1 if ψ is σ−symmetric, h− = 1 if ψ is σ−alternating. Hence

1 r(r + 1) r(r − 1) h1 − h1 = − P − + 1 + − 2 p∈R 2 2 = −nr + 1 if ψ is σ−symmetric. (2.2) 1 r(r + 1) r(r − 1) h1 − h1 = − P − − 1 + − 2 p∈R 2 2 = −nr − 1 if ψ is σ−alternating.

From (2.1) and (2.2), we deduce

r2 nr h1 = (g − 1) + if ψ est σ−symmetric. − 2 X 2 r2 nr h1 = (g − 1) − if ψ est σ−alternating. + 2 X 2 The other equalities are consequences of Hurwitz formula.

In particular, one deduces

(r + 2)(r − 1) dim(SU σ,+(r)) = (r2 − 1)(g − 1) + n , X Y 2 (r + 1)(r − 2) dim(SU σ,−(r)) = (r2 − 1)(g − 1) + n . X Y 2 Remark 2.5.3. Another method to compute the dimensions is to consider the map

1 ∗ 1 H (X,E ⊗ E ) → H (X, OX ).

This map is not equivariant with respect to the action of σ. In fact, the image by this map 1 ∗ 1 ∗ of H (E ⊗ E )− when E is σ−symmetric (resp. H (E ⊗ E )+ when E is σ−alternating) 1 is always included in H (X, OX )− (with respect to the canonical linearization on OX ).

Chapter 3

Hitchin systems

Hitchin in [Hit87] has defined and studied some integrable systems related to the moduli space of stable G−bundles over X, where G = GLr, Sp2m and SOr. Let MX (G) be this moduli space, the tangent space to MX (G) at a point [E] can be identified with 1 ∼ 0 ∗ H (X, Ad(E)) = H (X, Ad(E) ⊗ KX ) , where Ad(E) is the adjoint bundle associated to E, which is a bundle of Lie algebras isomorphic to g = Lie(G). By Serre duality, the fiber of the cotangent bundle is H0(X, Ad(E) ⊗ KX ). By considering a basis of the invariant polynomials under the adjoint action on g, one gets a map

k ∗ 0 M 0 di TEMX (G) = H (X, Ad(E) ⊗ KX ) −→ H (X,KX ), i=1 where the (di)i are the degrees of these invariant polynomials. Hitchin has shown that these two spaces have the same dimension. In the case G = GLr, a basis of the invariant polynomials is given by the coeﬃcients of the characteristic polynomial. If E is a stable vector bundle, then this gives rise to a map

r 0 M 0 i HE : H (X, End(E) ⊗ KX ) −→ H (X,KX ) =: W, i=1 which associates to each Higgs field φ, the coefficients of its characteristic polynomial. The associated map ∗ H : T MX (GLr) −→ W is called the Hitchin morphism. By choosing a basis of W , H is represented by d = 2 r (gX − 1) + 1 functions f1, . . . , fd. Hitchin has proved that this system is algebraically completely integrable, i.e. its generic fiber is an open set in an abelian variety of dimension d, and the vector fields Xf1 ,..., Xfd associated to f1, ··· , fd (defined using the canonical ∗ 2−form on T MX (GLr)) are linear. Moreover, let UX (r, 0) be the moduli space of stable vector bundles of rank r and degree 0 on X. Consider the map

∗ Π: T UX (r, 0) → UX (r, 0) × W whose ﬁrst factor is the canonical projection and the second factor is H . Then it is proved in [BNR89] that Π is dominant.

The main topic of this chapter is the study of the Hitchin systems for the anti-invariant and the invariant loci. We use these systems to identify the connected components of σ,+ σ,− UX (r) and UX (r). The irreducibility of the invariant locus (of a ﬁxed type) is already 42 Chapter 3. Hitchin systems know in more general setting (see [BS14]).

We stress that in this chapter we always assume, unless otherwise stated, that the vector bundles are stable.

3.1 Generalities on spectral curves and Hitchin systems

In this section we recall the general theory of spectral curves. Our main reference is [BNR89]. Let L be any line bundle over a smooth projective curve X. Consider the ruled surface over X given by −1 q¯ : S = P(OX ⊕ L ) → X, where for a vector bundle E we denote Sym•(E ) the symmetric algebra and • P(E ) = Proj(Sym (E )). −1 Hence a point in S lying over x ∈ X corresponds to a hyperplane in the ﬁber (OX ⊕ L )x. It follows that the total space of L denoted |L| is contained in S. ∼ Let O(1) be the relatively ample line bundle over S. It is well known thatq ¯∗O(1) = −1 OX ⊕ L . Hence O(1) has a canonical section, denoted by y, corresponding to the direct ∗ summand OX . Also by the projection formulaq ¯∗(¯q L ⊗ O(1)) is isomorphic to L ⊕ OX , so it has also a canonical section which we denote by x. Let r M 0 i s = (s1, ··· , sr) ∈ H (X,L ) =: WL i=1 be an r−tuple of global sections of Li and consider the global section r ∗ r−1 ∗ r 0 ∗ r x + (¯q s1)yx + ··· + (¯q sr)y ∈ H (S, q¯ KX ⊗ O(r)). (3.1)

We denote by X˜s its zero scheme which is a curve. We say that X˜s is the spectral curve associated to s ∈ WL. Denote q : X˜s → X the restriction ofq ¯ to X˜s. It is clear that X˜s is ﬁnite cover of degree r of X and its ﬁber over p ∈ X is given by the homogeneous equation 1 in P r r−1 r x + s1(p)x y ··· + sr(p)y = 0.

Lemma 3.1.1. The set of elements s ∈ WL corresponding to smooth spectral curves X˜s is open. In particular it is dense whenever it is not empty.

Proof. Assume that X˜s is integral (i.e. reduced and irreducible, which is true for general s ∈ W , see [BNR89] Remark 3.1) and let r r−1 P (x, t) = x + s1(t)x + ··· + sr(t) = 0 be the equation of X˜s locally over a point p ∈ X, where t is a local parameter near p. Then, by the Jacobian criterion of smoothness, X˜s is singular at a point λ ∈ X˜s over p if and only if ∂P ∂P (λ, 0) = (λ, 0) = 0, ∂x ∂t i.e. r−1 r−2 rλ + (r − 1)s1(0)λ + ··· + sr−1(0) = 0, 0 r−1 0 r−2 0 s1(0)λ + s2(0)λ + ··· + sr(0) = 0.

Clearly these two equations give a closed condition on s = (s1, ··· , sr) ∈ WL. Hence the set of s ∈ WL corresponding to smooth curves X˜s is open. 3.1. Generalities on spectral curves and Hitchin systems 43

Remark 3.1.2. We remark that the criterion of smoothness given in [BNR89] Remark 3.5, is not correct. In fact the criterion assumes that the singular point is located at λ = 0.

Remark 3.1.3. An alternative way to construction X˜s is as follows: consider the symmetric • −1 OX −algebra Sym (L ). Deﬁne the ideal * + M −r • −1 I = si(L ) ⊂ Sym (L ), i

0 i −r −r+i where si ∈ H (X,L ) is seen here as an embedding si : L → L . Then X˜s can be deﬁned as Spec Sym•(L−1)/I.

Suppose that X˜s is smooth and let S˜ = Ram(X˜s/X) ⊂ X˜s be the ramification divisor of q : X˜s → X. Recall that q O =∼ O ⊕ L−1 ⊕ · · · ⊕ L−(r−1), ∗ X˜s X hence, by duality of finite flat morphisms (see e.g. [Har77] Ex III.6.10) ∗ q O (S˜) =∼ q O =∼ O ⊕ L ⊕ · · · ⊕ Lr−1. ∗ X˜s ∗ X˜s X

In particular, using the fact that for any line bundle M over X˜s

det(q M) = det(q O ) ⊗ Nm (M), ∗ ∗ X˜s X˜s/X where Nm : Pic(X˜ ) → Pic(X) is the norm map, we deduce X˜s/X s

deg(S˜) = r(r − 1)deg(L).

Furthermore, by Hurwitz formula, we have K = q∗K (S˜). Thus, by the projection X˜s X formula we get q K =∼ K ⊕ K L ⊕ · · · ⊕ K Lr−1. ∗ X˜s X X X It follows that the genus g of X˜ is X˜s s r(r − 1) g ˜ = deg(L) + r(gX − 1) + 1. Xs 2

Recall that for a stable vector bundle E, the Hitchin map

0 ∗ HE : H (X,E ⊗ E ⊗ L) → WL is deﬁned by i ! i ^ s −→ HE(s) = (−1) Tr( s) , i where Tr is the trace map. We recall a very important result from [BNR89].

Proposition 3.1.4. Let X˜s be an integral (resp. smooth) spectral curve over X associated to s ∈ W . Then there is a one-to-one correspondence between torsion-free O −modules L X˜s of rank 1 (resp. Pic(X˜s)) and the isomorphism classes of pairs (E, φ) where E is a rank r vector bundle and φ : E → E ⊗ L is a morphism such that HE(φ) = s 44 Chapter 3. Hitchin systems

Maybe the most important case of spectral curves is when L = KX . We denote simply by W the space W . In this case, the genus g of X˜ is g = r2(g − 1) + 1, which KX X˜s s X˜s X coincides with the dimension of the moduli space UX (r, 0) of stable vector bundles of rank r and degree 0 over X. In [BNR89] it is proved that the map

∗ Π: T UX (r, 0) → UX (r, 0) × W is dominant. Moreover, the fiber H −1(s) of a general point s ∈ W is isomorphic to an m open subset of Pic (X˜s), where m = r(r − 1)(gX − 1). We claim that this is still true for the classical algebraic groups Sp2m et SOr. Consider the moduli spaces MX (Sp2m) and MX (SOr) of Sp2m−bundles and SOr−bundles respectively which are stable as vector bundles. Define m M 0 2i WSp2m = H (X,KX ), i=1 and ( Lr/2−1 H0(X,K2i) ⊕ H0(X,Kr/2) r ≡ 0 mod 2 W = i=1 X X . SOr L(r−1)/2 0 2i i=1 H (X,KX ) r ≡ 1 mod 2 ˜ ˜ For general s ∈ WSp2m the curve Xs is smooth, and for general s ∈ WSOr the associated Xs is nodal curve. In this case we denote Xˆs its normalisation. In both cases, the involution ˜ of the ruled surface S that sends x to −x induces an involution on Xs, we denote it by ι. Remark that in the singular case, ι lifts to an involution on Xˆ without fixed points. Recall that Hitchin ([Hit87]) has proved that the map Π induces maps

∗ T MX (Sp2m) −→ MX (Sp2m) × WSp2m , ∗ (3.2) T MX (SOr) −→ MX (SOr) × WSOr .

Moreover, the generic ﬁber in the case of symplectic bundles is isomorphic to an open set of a translate of the Prym variety of X˜s → X˜s/ι. In the case of orthogonal bundles, the generic ﬁber is an open dense of the Prym variety of Xˆs → Xˆs/ι. We refer to [Hit87] for more details.

Proposition 3.1.5. The restrictions of Π given in (3.2) are dominant. Moreover, for ˜ ˜ general s ∈ WSp2m (resp. s ∈ WSOr ), if P is a translation of the Prym variety of Xs → Xs/ι (resp. Xˆs → Xˆs/ι), then the pushforward map

P 99K MX (Sp2m)(resp. MX (SOr)) is dominant.

Proof. Laumon has proved in [Lau88] that the nilpotent cone

∗ ΛG ⊂ T MX (G) is Lagrangian, for any reductive algebraic group G. In particular, for G = Sp2m (resp. G = SOr), we deduce that the locus of G−bundles E such that

0 HE : H (X, Ad(E) ⊗ KX ) → WSp2m ( resp. WSOr ) is dominant, forms an open dense subset of MX (G). Indeed, we have

dim(ΛG) = dim(MX (G)), 3.2. The Hitchin system for anti-invariant vector bundles 45

∗ and the restriction of the canonical projection T MX (G) → MX (G) to ΛG is surjective (because (E, 0) ∈ ΛG for any G−bundle E). Hence by dimension theorem, it follows that there exists an open dense subset of MX (G) over which ΛG is reduced to the zero section ∗ of T MX (G). This open subset is by deﬁnition the set of very stable bundles E, for which, the map HE is dominant. It follows that the restrictions of Π given in (3.2) are dominant maps. Hence for general s ∈ WSp2m (resp. s ∈ WSOr ), we get a dominant maps

−1 H (s) −→ MX (Sp2m) (resp. MX (SOr)).

−1 Furthermore, if S is the ramiﬁcation of X˜s/ι → X (resp. Xˆs/ι → X), P = Nm (O(S)), where Nm is the norm map attached to the cover X˜s → X˜s/ι (resp. Xˆs → Xˆs/ι), then, by [Hit87], H −1(s) is an open dense of P. Thus the pushforward map

P 99K MX (Sp2m) (resp. MX (SOr)) is dominant rational map. Remark that in the symplectic case, the involution ι has some ﬁxed points, this implies that P is irreducible. While in the orthogonal case, ι is ´etale,hence P has two connected components, each one of them dominates a connected component of MX (SOr). In particular we deduce a cohomological criterion identifying the two connected components of MX (SOr). More explicitly, take an even theta characteristic κ of X, then the two components are distinguished by the parity of h0(X,E ⊗ κ). This is the same as the criterion given by the Stiefel-Whitney class (see for example [Bea06]).

3.2 The Hitchin system for anti-invariant vector bundles

For s ∈ W , we denote by q : X˜s → X the associated spectral cover of X, and by S˜ = Ram(X˜s/X) its ramiﬁcation divisor. Fix the positive linearizations on KX and OX (see Remark 1.1.3). Recall that this linearization equals id over the ramiﬁcation points. We denote these linearizations by

∗ ∗ η : σ KX → KX , ν : σ OX → OX .

i The linearization η induces an involution on the space of global sections of KX for each i 1. We deﬁne > r σ,+ M 0 i W = H (X,KX )+. i=1 σ,+ Proposition 3.2.1. Consider an r−tuple of global sections s = (s1, ··· , sr) ∈ W and let X˜s be the associated spectral curve over X. Then the involution σ : X → X lifts to an involution σ˜ on X˜s and O(S˜) descends to Y˜s := X˜s/σ˜. Proof. We have an isomorphism

tν⊗ tη −1 ∗ −1 OX ⊕ KX −−−−→ σ (OX ⊕ KX ), −1 ˜ −1 which induces an involutionσ ¯ on S = P(OX ⊕ KX ). Let Xs ⊂ P(OX ⊕ KX ) be the spectral curve associated to s. ∼ Recall that the canonical section y of O(1) is identiﬁed with the identity section ofq ¯∗O(1) = −1 OX ⊕ KX , therefore it isσ ¯−invariant. The section x is by deﬁnition the canonical section ∗ ∗ ofq ¯ KX ⊗O(1). In fact it can be seen as the canonical section ofq ¯ KX → |KX |, where |KX | 46 Chapter 3. Hitchin systems

is the total space of KX . Hence x is invariant with respect to the positive linearization. ⊗k ∗ As by deﬁnition η (σ (sk)) = sk, we deduce that ∗ k r−k ∗ k r−k σ¯((¯q sk)y x ) = (¯q sk)y x .

Thus the section defining X˜s r ∗ r−1 ∗ r 0 ∗ r x + (¯q s1)yx + ··· + (¯q sr)y ∈ H (S, q¯ KX ⊗ O(r)) isσ ¯−invariant. Henceσ ¯(X˜s) = X˜s, soσ ¯ induces an involution on X˜s which we denote by σ˜. Remark thatσ ¯ acts trivially on the fibers ofq ¯ : S → X over the ramification points of π : X → Y . Thus the ramification locus ofσ ˜ is q−1(R).

By Hurwitz formula we have O(S˜) = K ⊗ q∗K−1. We also know by Lemma 1.1.2 X˜s X that K (resp. K ) descends to Y˜ (resp. Y ). Moreover, K =π ˜∗K (R˜) (resp. K = X˜s X s X˜s Y˜ X ∗ π KY (R)), where R˜ = Ram(X˜s/Y˜s), and we have used the notation of the commutative diagram q X˜s / X

π˜ π q˜ Y˜s / Y, since O(R˜) = q∗O(R), it follows that

O(S˜) = K ⊗ q∗K−1 X˜s X =π ˜∗K ⊗ q∗(π∗K−1) ⊗ O(R˜) ⊗ q∗O(−R) Y˜s Y =π ˜∗ K ⊗ q˜∗K−1 . Y˜s Y

Since by Hurwitz formula K ⊗ q˜∗K−1 = O(S), where S = Ram(Y˜ /Y ), we deduce that Y˜s Y s O(S˜) =π ˜∗O(S).

We keep the notations of the last proposition hereafter. σ,+ Remark 3.2.2. Remark that for s ∈ W , Y˜s is a spectral cover of Y associated to some spectral data of the line bundle L = KY ⊗∆ over Y . This is because the sections si descend to Y . Lemma 3.2.3. Let F be a σ−linearized vector bundle, and consider the positive linearization on KX . Then the Serre duality isomorphism 1 ∗ ∼ 0 ∗ H (X,F ) −→ H (X,F ⊗ KX ) is anti-equivariant with respect to the induced involutions on the two spaces. Proof. If F is a σ−linearized vector bundle, we have an equivariant perfect pairing:

0 1 ∗ 1 ∼ H (X,F ) ⊗ H (X,F ⊗ KX ) → H (X,KX ) −→ C. As the ﬁxed linearization is the positive one, it follows by Remark 1.1.3 that

1 1 ∗ H (X,KX )− = H (X, π (KY ⊗ ∆))− 1 −1 = H (Y,KY ⊗ ∆ ⊗ ∆ ) 1 = H (Y,KY ) = C. 3.2. The Hitchin system for anti-invariant vector bundles 47

So 1 1 H (X,KX ) = H (X,KX )−. Since the above pairing is equivariant, we get the result.

Let ς be the canonical linearization on E ⊗ σ∗E given by the transposition, then the ∗ 0 ∗ linearization ς ⊗η on E ⊗σ E ⊗KX induces an involution on H (X,E ⊗σ E ⊗KX ) which we denote by f. By the above proposition, one gets an isomorphism

∗ σ,+ ∼ 0 ∗ T UX (r) −→ H (X,E ⊗ σ E ⊗ KX )+, ∗ σ,− ∼ 0 ∗ T UX (r) −→ H (X,E ⊗ σ E ⊗ KX )−,

th We denote by Hi the i component of the Hitchin map

0 ∗ HE : H (X,E ⊗ σ E ⊗ KX ) → W.

Proposition 3.2.4. Let E be σ−anti-invariant stable vector bundle and ψ : σ∗E =∼ E∗ be an isomorphism.

1. If ψ is σ−symmetric, then Hi induces a map

0 ∗ 0 i Hi : H (X,E ⊗ σ E ⊗ KX )+ → H (X,KX )+.

2. If ψ is σ−alternating, then Hi induces a map

0 ∗ 0 i Hi : H (X,E ⊗ σ E ⊗ KX )− → H (X,KX )+.

0 ∗ 0 Proof. Let f be the involution on H (X,E ⊗σ E ⊗KX ) defined above. Let φ ∈ H (X,E ⊗ ∗ P ∗ σ E ⊗ KX ), locally we can write φ = k sk ⊗ σ (tk) ⊗ αk, where αk (resp. sk, tk) are local sections of KX (resp. E). We can see the section φ as a map E → E ⊗ KX which is defined locally by X ∗ x −→ φ(x) = hψ(σ (tk)), xi sk ⊗ αk. k Thus Vi φ is defined locally by

i ^ φ(x1 ∧ · · · ∧ xi) = i!φ(x1) ∧ · · · ∧ φ(xi)     X ∗ X ∗ = i!  hψ(σ (tk1 )), x1i sk1 ⊗ αk1  ∧ · · · ∧  hψ(σ (tki )), xii ski ⊗ αki  k1 ki  i  X ∗ ∗ O = i! hψ(σ (tk1 )), x1i · · · hψ(σ (tki )), xii sk1 ∧ · · · ∧ ski ⊗  αkj  k1,...,ki j=1  i  X O = i! det ψ(σ∗(t )), x s ∧ · · · ∧ s ⊗ α kj l j,l k1 ki  kj  k1<···

For the last equality, we use the canonical isomorphism Vk E∗ =∼ (Vk E)∗ given by the determinant. It follows that (locally) we have

i i ^ X ∗ ∗ O φ = i! sk1 ∧ · · · ∧ ski ⊗ σ (tk1 ) ∧ · · · ∧ σ (tki ) ⊗ αkj . k1<···

hψ(σ∗(t)), si = ν(σ∗ hψ(σ∗(s)), ti).

Hence

i i t ∗ ^ Hi(f(φ)) = (−1) Tr( (σ ( φ))) * i + i i X ^ ∗ ∗ O ∗ = (−1) i! ψ(σ (sk1 ) ∧ · · · ∧ σ (ski )), tk1 ∧ · · · ∧ tki η(σ (αkj )) k1<···

⊗i ∗ Thus, if f(φ) = φ, then η (σ (Hi(φ))) = Hi(φ). 2. If ψ is σ−alternating, so we have

hψ(σ∗(t)), si = −ν(σ∗ hψ(σ∗(s)), ti).

By the above calculation, it follows that

i ⊗i ∗ Hi(f(φ)) = (−1) η (σ (Hi(φ))).

On the other hand, it is clear that

i Hi(−φ) = (−1) Hi(φ),

so if f(φ) = −φ, then ⊗i ∗ η (σ (Hi(φ))) = Hi(φ).

σ,+ σ,+ We claim that dim(W ) = dim(UX (r)). Indeed we have 0 i ∼ 0 i H (X,KX ) = H (Y, π∗KX ) 0 i i 0 i−1 = H (Y,KY ⊗ ∆ ) ⊕ H (Y,KY ⊗ ∆ ).

As the ﬁxed linearization on KX is the positive one, by Remark 1.1.3, we obtain 0 i ∼ 0 i i H (X,KX )+ = H (Y,KY ⊗ ∆ ), hence 0 i h (X,KX )+ = (2i − 1)(gY − 1) + in. 3.2. The Hitchin system for anti-invariant vector bundles 49

It follows that r σ,+ X dim(W ) = (2i − 1)(gY − 1) + in i=1 r(r + 1) = r2(g − 1) + n. Y 2 Remark 3.2.5. We use Riemann-Roch and Lefschetz ﬁxed point theorem (see AppendixF) to calculate the dimension of W σ,+ by a second method. Two cases should be distinguished 0 0 i (for simplicity of notations, we denote by h±(i) the dimensions of H (KX )±).

1. i = 1: as the Serre duality is anti-equivariant for the positive linearization on KX , 1 0 we deduce that h (X,KX )− = h (X, OX )+ = 1 (ree remark 3.2.3). Hence

 0 0 h+(1) + h−(1) = gX 1 h0 (1) − h0 (1) = −1 + P2n 1 = n − 1.  + − 2 k=1

0 So we get h+(1) = gY − 1 + n.

1 i 2. i > 2: in this case H (X,KX ) = 0, it follows

 0 0 h+(i) + h−(i) = (2i − 1)(gX − 1) 1 h0 (i) − h0 (i) = P2n 1 = n.  + − 2 k=1 1 Hence h0 (i) = ((2i − 1)(g − 1) + n) = (2i − 1)(g − 1) + in. + 2 X Y Finally, we get r X r(r + 1) h0(X,Ki ) = r2(g − 1) + n. X + Y 2 i=1 σ,+ σ,− To study the irreducibility of UX (r) and UX (r), we will use the notion of very stable vector bundles, which has been introduced in [Lau88]. Let E be a stable vector bundle, and let φ : E → E ⊗ KX be a Higgs ﬁeld. We say that φ is nilpotent if the composition of the maps φ φ⊗id 2 r−1 φ⊗id r E −→ E ⊗ KX −→ E ⊗ KX → · · · → E ⊗ KX −→ E ⊗ KX is identically zero.

Deﬁnition 3.2.6. We say that a vector bundle E is very stable if E has no nilpotent Higgs ﬁeld other than 0.

If E is a very stable vector bundle, then the Hitchin morphism

0 ∗ HE : H (X,E ⊗ E ⊗ KX ) → W

−1 is dominant. Indeed, by deﬁnition, HE (0) = {0}, but the two spaces have the same dimension, this implies that HE is dominant. One of the main results of [Lau88] is that the locus of very stable vector bundles is an open dense subscheme of the moduli space of vector bundles. 50 Chapter 3. Hitchin systems

Deﬁnition 3.2.7. We say that a σ−symmetric (resp. σ−alternating) anti-invariant vector bundle E is very stable if E has no nilpotent Higgs ﬁeld

0 ∗ 0 ∗ φ ∈ H (E ⊗ σ E ⊗ KX )+ ( resp. φ ∈ H (E ⊗ σ E ⊗ KX )−) other than 0.

∗ Let T UX (r, 0) be the cotangent bundle of UX (r, 0). This bundle is invariant with ∗ ∗ respect to the involution E → σ E on UX (r, 0). In fact, this is true more generally for any variety Z with an involution τ. To see this consider the diﬀerential of τ, it gives a linear isomorphism dτ : TZ −→ τ ∗TZ, 2 ∗ but τ = idZ , this implies that dτ ◦ τ dτ = id. Thus dτ is a linearization on TZ, hence tdτ is a linearization on T ∗Z. ∗ ∗ In particular, in our case, the involution E → σ E of UX (r, 0) lifts to an involution on ∗ ∗ ∼ 0 ∗ ∗ ∼ ∗ T UX (r, 0). If we identify TEUX (r, 0) = H (X,E ⊗ σ E ⊗ KX ) using ψ : σ E = E 0 ∗ and Serre duality, then this lifting is the involution f on H (X,E ⊗ σ E ⊗ KX ) given by f(φ) = t(σ∗φ) in the σ−symmetric case, and f(φ) = − t(σ∗φ) in the σ−alternating case. ∗ σ,+ Moreover, by section 2.5, the ﬁxed locus of this involution is the cotangent bundle T UX (r) ∗ σ,− ∗ σ,± ∗ (resp. T UX (r)). Hence we can consider both T UX (r) as closed subspaces of T UX (r, 0). ∗ Moreover, the tautological symplectic form on T UX (r, 0) restricts to the tautological forms ∗ σ,± on T UX (r).

∗ Following the notations of [Lau88], let ΛX,r ⊂ T UX (r, 0) be the nilpotent cone, that’s the set of (E, φ) with φ nilpotent Higgs ﬁeld. Set

σ,+ ∗ σ,+ σ,− ∗ σ,− ΛX,r = ΛX,r ∩ T UX (r) , ΛX,r = ΛX,r ∩ T UX (r).

σ,+ σ,− ∗ σ,+ Theorem 3.2.8. The nilpotent cone ΛX,r (resp. ΛX,r) is Lagrangian in T UX (r) (resp. ∗ σ,− T UX (r)). In particular the locus of very stable anti-invariant vector bundles is dense in σ,+ σ,− UX (r) (resp. UX (r)). Proof. We prove the σ−symmetric case, the σ−alternating is absolutely the same. If V is symplectic space, then the restriction of a Lagrangian subspace L ⊂ V to a symplectic σ,+ subspace F ⊂ V is an isotropic subspace of F , this implies that ΛX,r is an isotropic ∗ σ,+ σ,+ subspace of T UX (r). In particular its dimension is at most dim(UX (r)). But it is σ,+ σ,+ clear that UX (r) ⊂ ΛX,r, by seeing any anti-invariant vector bundle E as the trivial pair σ,+ (E, 0) ∈ ΛX,r. This implies that 1 dim(Λσ,+) = dim(T ∗U σ,+(r)), X,r 2 X σ,+ ∗ σ,+ hence ΛX,r is Lagrangian of T UX (r).

3.2.1 σ−symmetric case The ramiﬁed case

Suppose that π : X → Y is ramiﬁed and denote m = r(r − 1)(gX − 1). Recall that deg(S˜) = 2m, where S˜ = Ram(X˜s/X). We ﬁx the positive linearization on O(S˜). σ,+ + m For general s ∈ W , consider the subvariety P ⊂ Pic (X˜s) of isomorphism classes of line bundles L such that Nm(L) =∼ O (S), g Y˜s 3.2. The Hitchin system for anti-invariant vector bundles 51

˜ 0 ˜ 0 ˜ where S = Ram(Ys/Y ), and Nmg : Pic (Xs) → Pic (Ys) the norm map attached toπ ˜ : X˜s → Y˜s. By Proposition 3.2.1,π ˜∗O (S) = O (S˜), it follows that for each L ∈ P+, we have Y˜s X˜s

σ˜∗L =∼ L−1(S˜).

In particular any M ∈ P+ gives by tensor product an isomorphism

+ ∼ P −→ Prym(X˜s/Y˜s).

Lemma 3.2.9. For general s ∈ W σ,+, we have

+ σ,+ dim(P ) = dim(UX (r)). Proof. For general s ∈ W σ,+, the curve X˜ is smooth and its genus is g = r2(g − 1) + 1. s X˜s X Indeed, by Lemma 3.1.1, and because W σ,+ is irreducible, it suffices to prove that there σ,+ ˜ 0 r exists an s ∈ W such that Xs is smooth. To do so, take sr ∈ H (X,KX )+ to be a general section that has just simple roots which are different from the ramification points (i.e. outside a finite union of hyperplanes of sections vanishing at points of R). This is possible 0 r 0 r 0 r because the hyperplane H (KX (−p)) ⊂ H (KX ) contains H (KX )−, so necessarily it does 0 r not contain H (KX )+, and this for every p ∈ R. Then using the proof of Proposition 3.1.1, we deduce that the spectral curve attached to s = (0, ··· , 0, sr) is smooth. Moreover, if X˜s is smooth then, by Lemma 1.2.2, we deduce that R˜ = Ram(X˜s/Y˜s) = q−1(R) has no multiple points (i.e. reduced divisor). Furthermore we have deg(R˜) = 2rn, so we get

dim(P+) = g − g X˜s Y˜s 1 = (g ˜ − 1 + rn) (by Riemann-Roch) 2 Xs r2 rn = (g − 1) + . 2 X 2

Remark 3.2.10. We can prove directly that if a ramiﬁcation point in R˜ is double then it is singular point. Indeed, assume that a ∈ R˜ is a multiple point over p ∈ R, and let t be a local parameter in a neighborhood of p. Then X˜s is given locally over p by the equation

r r−1 P (x, t) = x + s1(t)x + ··· + sr(t) = 0.

In particular, because a is multiple point, we have ∂P P (a, 0) = (a, 0) = 0. ∂x 0 i Now, as si ∈ H (X,KX )+, it follows that si(−t) = si(t), so for each i, we can write locally near p 2 si(t) = a0 + t a2 + ··· . 2 This implies that P (x, t) = P0(x) + t P2(x) + ··· , where Pi are polynomials in x. Hence ∂P (a, 0) = 0. ∂t

By the Jacobian criterion of smoothness, we deduce that X˜s is singular at a. 52 Chapter 3. Hitchin systems

+ Remark 3.2.11. We can calculate dim(P ) using the fact that Y˜s is spectral curve over Y . Indeed, since the s1, . . . , sr descend to Y , we see that Y˜s is a spectral curve over Y with respect to the line bundle KY ⊗ ∆. Hence by subsection 3.1, we deduce that r(r − 1) g ˜ = deg(KY ⊗ ∆) + r(gY − 1) + 1 Ys 2 r(r − 1) = (2g − 2 + n) + r(g − 1) + 1 Y 2 Y r(r − 1) = r2(g − 1) + n + 1. Y 2 It follows that

dim(P+) = g − g X˜s Y˜s r(r − 1) = r2(g − 1) + 1 − r2(g − 1) + n + 1 X Y 2 r(r + 1) = r2(g − 1) + n. Y 2

Theorem 3.2.12. Suppose that π : X → Y is ramified. Then for general s ∈ W σ,+, the rational pushforward map + σ,+ q∗ : P 99K UX (r) σ,+ is dominant. In particular UX (r) is irreducible. Proof. First, by the duality for finite flat morphisms, this map is well defined, more precisely, one has

∗ ∼ ∗ σ q∗L = q∗(˜σ L) ∼ −1 = q∗(L ⊗ O(S˜)) ∼ ∗ = (q∗L) , the last isomorphism is the duality for finite flat morphisms (see for example [Har77] ∗ ∗ Ex. III.6.10). The isomorphism ψ : σ (q∗L) → (q∗L) is defined using the pairing ψ˜ : ∗ 0 0 −1 q∗L ⊗ σ (q∗L) → OX , which is defined as follows: for v, w ∈ H (U, q∗L) = H (q (U),L), we put ψ˜(v ⊗ σ˜∗w) · ξ = hv, σ˜∗wi , 0 where ξ ∈ H (X˜s, O(S˜)) is the canonical section equals the derivative of q : X˜s → X, and h, i : L ⊗ σ˜∗L → O(S˜) is an isomorphism. This last isomorphism isσ ˜−symmetric for the positive linearization on O(S˜). Indeed, it is a global section ofσ ˜∗L−1 ⊗ L−1 ⊗ O(S˜) (=∼ O ) and because we fixed the positive linearization on O(S˜) we deduce X˜s

0 ˜ ∗ −1 −1 ˜ 0 ˜ ∗ −1 ∗ H (Xs, σ˜ L ⊗ L ⊗ O(S))+ = H (Xs, π˜ Nm(g L ) ⊗ π˜ (O(S)))+ 0 ∗ ∗ = H (X˜s, π˜ (O(−S)) ⊗ π˜ (O(S)))+ = H0(Y˜ , O ) s Y˜s = C.

Further ξ isσ ˜−invariant global section of O(S˜) with respect to the positive linearization. Hence ψ is σ−symmetric. 3.2. The Hitchin system for anti-invariant vector bundles 53

Conversely, giving a σ−symmetric anti-invariant stable vector bundle E and φ ∈ 0 ∗ H (X,E ⊗ σ E ⊗ KX )+ such that HE(φ) = s, then the corresponding line bundle over X˜s is in P+. To see this, consider the exact sequence (see [BNR89] Remark 3.7)

∗ ∗ q φ−x ∗ ∗ 0 → L(−S˜) −→ q (E) −−−−→ q (E ⊗ KX ) −→ L ⊗ q KX → 0. (3.3)

∗ ∗ By taking the dual, pulling back byσ ˜ and than taking the tensor product byσ ˜ q KX , we get the exact sequence

∗ t ∗ ∗ ∗ −1 ∗ ∗ ∗ σ˜ ( (q φ))−σ˜ x ∗ ∗ ∗ ∗ −1 ∗ 0 → σ˜ (L ) → σ˜ q (E ) −−−−−−−−−−→ σ˜ (q (E ⊗ KX )) → σ˜ (L (S˜) ⊗ q KX ) → 0. (3.4) Since φ is invariant, i.e. t(σ∗φ) = φ, the middle maps of the exact sequences (3.3) and (3.4) are identiﬁed using the isomorphism ψ : σ∗E → E∗, hence they have isomorphic kernels. This implies that L(−S˜) =∼ σ˜∗(L−1). Thus L ∈ P+.

+ Moreover, by Lemma 3.2.9 we deduce that whenever X˜s is smooth we have dim(P ) = σ,+ dim(UX (r)). σ,+ Now, if E ∈ UX (r) is very stable then the map

0 ∗ σ,+ HE : H (X,E ⊗ σ E ⊗ KX )+ → W is dominant, it follows that the map

∗ σ,+ σ,+ σ,+ Π: T UX (r) −→ UX (r) × W

σ,+ is dominant too, because the locus of very stable vector bundles is dense inside UX (r) by Theorem 3.2.8. In particular, ﬁxing a general s ∈ W σ,+, we obtain a dominant morphism

−1 σ,+ H (s) −→ UX (r),

∗ σ,+ σ,+ where H : T UX (r) → W is the Hitchin morphism. But, by Proposition 3.1.4 and −1 ∼ what we have said above, we deduce that (E, φ) ∈ H (s) if and only if E = q∗L for some L ∈ P+. It follows that the rational map

+ σ,+ q∗ : P 99K UX (r) is dominant. −1 As R˜ = q (R), one deduces that X˜s → Y˜s is ramiﬁed. This implies the connectedness + σ,+ of P , hence the irreducibility of UX (r).

The ´etalecase Assume that the cover π : X → Y is ´etale.In this case, any σ−invariant vector bundle over X descends to Y by Kempf’s Lemma. In particular, we have

∗ ∗ KX = π (KY ⊗ ∆) = π KY .

Recall that the linearization on KX attached to KY ⊗ ∆ is called the positive linearization and that Serre duality is anti-equivariant with respect to this linearization. Remark that O(S˜) =π ˜∗O(S) =π ˜∗(O(S) ⊗ ∆),˜ where ∆˜ = det(˜π O )−1. We ﬁx the ∗ X˜s linearization on O(S˜) attached to the O(S) and we continue calling it the positive linearization. 54 Chapter 3. Hitchin systems

Theorem 3.2.13. Suppose that π : X → Y is ´etale,then the pushforward rational map q∗ induces a dominant map + σ,+ q∗ : P 99K UX (r). σ,+ In particular UX (r) has two connected components. + Proof. Clearly X˜s → Y˜s is ´etaleif and only if X → Y is. Hence P has two connected components. We show that it is impossible to produce the same stable vector bundle E as the direct image of two line bundles from the two connected components of P+. To see this assume that we have L and L0, two line bundles each from a connected component + ∼ 0 σ,+ ˜ of P , such that q∗L = q∗L ∈ UX (r). Let M be a line bundle on Xs such that M ˜ 2 ∼ ˜ descends to Ys and Nm(g M) = O(S), in particular M = O(S). Let κ be an even theta −1 0 characteristic on X˜s such that M ⊗κ is the pullback of a theta characteristic κ on Y , i.e M −1 ⊗ κ = q∗ (π∗(κ0)), note that such a pair (M, κ) exists by Lemma 3.2.14 below. Then, by [BL04] Theorem 12.6.2, we know that

0 −1 0 0 −1 h (X˜s,L ⊗ M ⊗ κ) ≡ 0 mod 2, h (X˜s,L ⊗ M ⊗ κ) ≡ 1 mod 2 Using the projection formula, this gives

0 ∗ 0 0 0 ∗ 0 h (X, q∗L ⊗ π κ ) ≡ 0 mod 2 and h (X, q∗L ⊗ π κ ) ≡ 1 mod 2 a contradiction. σ,+ ˜ Moreover, if E ∈ UX (r), then the associated line bundle L over Xs constructed in the ∗ ∼ −1 proof of Theorem 3.2.12, verifiesσ ˜ L = L (S˜). Sinceπ ˜ : X˜s → Y˜s is étale,it follows that either Nm(L) = O (S), or Nm(L) = O (S) ⊗ ∆.˜ But ψ induces aσ ˜−symmetric isomor- g Y˜s g Y˜s phismσ ˜∗L → L−1(S˜), and because we have fixed the positive linearization on O (S˜), it X˜s follows that Nm(L) = O (S). So L ∈ P+. g Y˜s + σ,+ Now the image of the rational map q∗ : P → UX (r) has two connected components, which are dense, and by Mumford [Mum71], the map

0 0 E → h (X, π∗E ⊗ κ ) mod 2 σ,+ is constant under deformation of E. Hence UX (r) can’t be irreducible. It follows that it has two connected components.

Lemma 3.2.14. Suppose that π : X → Y is ´etaleand X˜s is smooth. Then there exist an even theta characteristic κ on X˜s, and a line bundle M that descends to Y˜s and veriﬁes M 2 =∼ O(S˜), such that M −1 ⊗ κ descends to a theta characteristic on Y . −1 Proof. Recall that we denoted by ∆˜ := det π˜ O , note that ∆˜ is non-trivial 2−torsion ∗ X˜s line bundle over Y˜s. By [BL04] page 382, we know that there exists an even theta charac- 00 teristic, say κ , on Y˜s such that h0(κ00) ≡ h0(κ00 ⊗ ∆)˜ ≡ 0 mod 2.

If we set κ =π ˜∗κ00, we get by the projection formula

h0(κ) = h0(κ00) + h0(κ00 ⊗ ∆)˜ ≡ 0 mod 2, hence κ is even theta characteristic. Moreover, let N be a line bundle on Y˜s such that 2 ∼ N = O(S) (recall that S := Ram(Y˜s/Y ) has an even degree), then by Hurwitz formula, we have (N −1 ⊗ κ00)2 = (˜q∗κ0)2, 3.2. The Hitchin system for anti-invariant vector bundles 55 where κ0 is a (any) theta characteristic on Y . It follows that there exists a 2−torsion line bundle α on Y˜s such that N −1 ⊗ κ00 ⊗ α =q ˜∗κ0. It suﬃces to take M =π ˜∗(N ⊗ α−1).

σ,+ −1 Remark 3.2.15. The determinant induces a morphism det : UX (r) → P = Nm (OY ). σ,+ Note that P = UX (1). The composition of this map with the direct image q∗ gives a map P+ −→ P.

Moreover, each connected component of P+ dominates a connected component of P (it is even surjective). Indeed, let L ∈ P+ and M be a line bundle in P+ such that Nm (M) = δ, where δ = (q O )−1. Then L⊗M −1 is in the Prym variety of X˜ → Y˜ , X˜s/X ∗ X˜s s s hence can be written asσ ˜∗λ ⊗ λ−1, it follows that

det(q L) = Nm (L) ⊗ δ−1 = σ∗Nm (λ) ⊗ Nm (λ)−1. ∗ X˜s/X X˜s/X X˜s/X Then it suﬃces to recall that the image of the map λ −→ σ˜∗λ ⊗ λ−1 equals the identity component of the Prym variety when λ runs Pic0(X˜) and equals the other component when it runs Pic1(X˜). In particular, using Proposition 2.3.3, we deduce that for general line bundle L in both + connected components of P , q∗L is stable.

Trivial determinant case In this section, nothing is assumed on the cover π : X → Y , i.e. it may be ramiﬁed + −1 −1 σ,+ or not. Denote by Q = Nm (δ), where δ = (det(q∗O ˜ )) . For general s ∈ W , X˜s/X Xs + Q is isomorphic to the Prym variety of the spectral cover X˜s → X. In particular it is connected. Proposition 3.2.16. For general s ∈ W σ,+, P+ ∩ Q+ is connected. Proof. Fixing an element in P+ ∩ Q+ gives by tensor product an isomorphism

P ∩ Q −→∼ P+ ∩ Q+.

So it is suﬃcient to prove the connectedness of P ∩ Q (recall that P and Q are the Prym varieties ofπ ˜ : X˜ → Y˜ and q : X˜ → X respectively). The norm map Nm : J → J s s s g X˜s Y˜s induces a homomorphism ϑ : Q → Q, ˜ which is just the restriction of Nmg to Q, here Q is the Prym variety ofq ˜ : Ys → Y . We have a commutative diagram

∼ Q / J / J / / Q X˜s dX˜s b O O O O µ ∗ ˆ π˜ Nmgd ϑ ∼ Q / J / J / / Q, Y˜s cY˜s b where Qb and Qb are the dual abelian varieties of Q and Q respectively and µ : Q → Q is the morphism deﬁned by the factorization µ π˜∗| : Q −→ Q ,→ J . Q X˜s 56 Chapter 3. Hitchin systems

We obtain the commutative diagram

ϕQ Q / Qb (3.5) O O µ ϑˆ

ϕQ Q / Qb , where ϕQ : Q → Qb (resp. ϕQ : Q → Qb ) is the restriction of the principal polarization of J (resp. J ) to Q (resp. Q). By [BNR89] Remark 2.7, as the spectral covers are always X˜s Y˜s ramiﬁed, the types of these two polarizations are (1,..., 1, r, . . . , r) and (1,..., 1, r, . . . , r) | {z } | {z } gX gY respectively. Hence the degree of these two restrictions is r2gX and r2gY respectively.

Assume that X → Y is ramiﬁed. Then µ is injective. By Diagram (3.5) it follows that ˆ card(Ker(ϕQ ◦ µ)) = card(Ker(ϕQ ◦ ϑ)).

2g ∗ ∗ It is easy to see that card(Ker(ϕQ ◦µ)) = r Y . Indeed if L ∈ Ker(ϕQ ◦µ), thenπ ˜ L = q M for some M ∈ JX [r], but this implies that M descends to Y (here we use Kempf’s Lemma ∗ ∗ (1.1.1) to prove that if q M descends to Y˜s then M descends to Y ). Hence L =q ˜ N for ∗ ∗ some N ∈ JY [r], i.e. Ker(ϕQ ◦ µ) =q ˜ JY [r]. Asq ˜ is injective (c.f. [BNR89] Remark 3.10) this implies the result. ∗ 2gY ˆ But we also have Ker(ϕQ) =q ˜ JY [r], so card(Ker(ϕQ)) = r . This proves that ϑ is injective. By general theory of abelian varieties (see for example [BL04] Proposition 2.4.3), Ker(f) and Ker (fˆ) have the same number of connected components for any surjective morphism f : A → B between abelian varieties. Since ϑ is clearly surjective, it follows that Ker(ϑ) is connected, and by deﬁnition, the kernel of ϑ is precisely P ∩ Q.

If X → Y is ´etale,then so isπ ˜ : X˜s → Y˜s. In this case µ has degree 2. Let L ∈ ∗ ∗ Ker(ϕQ ◦ µ), then as aboveπ ˜ L = q M, for some M ∈ JX [r], hence M descends to Y , ∗ r r r say M = π N, as M = OX , we get N = OY or N = ∆, recall that ∆ is the 2−torsion line bundle attached to X → Y . Denote the set of rth roots of ∆ by T . It follows that L =q ˜∗N or L =q ˜∗N ⊗ ∆˜ for N ∈ J [r] ∪ T , where ∆˜ ∈ J [2] is the line bundle attached Y Y˜s ∗ to X˜s → Y˜s. Note that ∆˜ =q ˜ ∆. Since L ∈ Q, we deduce

∗ • if r is even, then multiplication by ∆ is an involution of JY [r] and T , so L ∈ q˜ JY [r] ∗ ∗ (becauseq ˜ T ∩Q = ∅). This implies that Ker(ϕQ ◦µ) =q ˜ JY [r], hence card(Ker(ϕQ ◦ µ)) = r2gY . It follows that ϑˆ is injective. So P ∩ Q is connected.

∼ ∗ • if r is odd, then multiplication by ∆ is an isomorphism JY [r] = T , and since q ∆ = ∆˜ we can assume that N ∈ J [r]. As Nm (˜q∗N ⊗∆)˜ = ∆r = ∆, thenq ˜∗N ⊗∆˜ 6∈ Q. Y Y˜s/Y ∗ 2g It follows L ∈ q˜ JY [r], hence card(Ker(ϕQ ◦ µ)) = r Y , and so we deduce again the connectedness of P ∩ Q.

Theorem 3.2.17. The pushforward map

+ + σ,+ q∗ : P ∩ Q 99K SU X (r)

σ,+ is dominant. In particular SU X (r) is irreducible. 3.2. The Hitchin system for anti-invariant vector bundles 57

Proof. Let P0 be the identity component of the Prym variety of X → Y , then it is clear that the map σ,+ σ,+ SU X (r) × P0 −→ UX,0 (r) σ,+ σ,+ is surjective, where UX,0 (r) is the connected component of UX (r) which is over P0 by the determinant map. It follows by Theorems 3.2.12 and 3.2.13 (in the ramiﬁed and σ,+ + ´etalecases respectively) that for general E ∈ UX,0 (r), there exists L ∈ P such that th −1 q∗L = E. Let λ ∈ P0 be an r root of det(E) . It follows by the projection formula that q (L ⊗ q∗λ) = E ⊗ λ ∈ SU σ,+(r). Note that L ⊗ q∗λ ∈ P+ because Nm(q∗λ) = O . Hence ∗ X g Y˜s σ,+ + a general E ∈ SU X (r) can be written as a direct image of some L ∈ P . But since det(q L) = δ−1 ⊗ Nm (L), ∗ X˜s/X where δ = det(q O )−1, we deduce that if q L has trivial determinant then Nm (L) = ∗ X˜s ∗ X˜s/X δ, thus L ∈ Q+. So we get a dominant rational map

+ + σ,+ P ∩ Q −→ SU X (r).

+ + σ,+ Now by Proposition 3.2.16, P ∩ Q is connected. So SU X (r) is irreducible. Remark 3.2.18. Remark that the map

σ,+ σ,+ SU X (r) × P −→ UX (r) is surjective, unless π : X → Y is étaleand r is even, for which its image is one connected component. Indeed, the ramified case is clear, so assume that π is étale,then if r is odd, the map [r]: P → P is surjective, and if r is even, its image is the identity component P0 ⊂ P . Now use Theorem 3.2.13 to deduce the result.

3.2.2 σ−alternating case The ramiﬁed case Suppose that π : X → Y is ramiﬁed and r is even. Let E be a σ−alternating stable 0 ∗ vector bundle. Consider the involution f on the space H (X,E ⊗ σ E ⊗ KX ) associated ∗ ∗ to the linearization ς ⊗ η on E ⊗ σ E ⊗ KX , where ς is the linearization on E ⊗ σ E equals the transposition. 0 ∗ ∗ ∼ ∗ Let φ ∈ H (X,E ⊗ σ E ⊗ KX )−. Using the isomorphism ψ : σ E = E , we can see φ as a t map E → E ⊗ KX (in fact we just identify φ and φ ◦ ( ψ) to simplify the notations). Then we have

Lemma 3.2.19. The following diagram

σ∗φ : σ∗E / σ∗E ⊗ σ∗K O X ψ−1 ψ⊗η

t ∗ ∗ φ : E / E ⊗ KX , commutes. 58 Chapter 3. Hitchin systems

Proof. Write (locally) X ∗ φ = sk ⊗ σ tk ⊗ αk, k and let v be a local section of E∗, then we have

! ∗ −1 X ∗ −1 ∗ ∗ (ψ ⊗ η) ◦ (σ φ) ◦ ψ (v) = (ψ ⊗ η) (σ ψ)(tk), ψ (v) σ sk ⊗ σ αk k ! X t −1 ∗ ∗ = (ψ ⊗ η) − ( ψ)(tk), ψ (v) σ sk ⊗ σ αk k ! X ∗ ∗ = −(ψ ⊗ η) htk, vi σ sk ⊗ σ αk k X ∗ ∗ = − htk, vi ψ(σ sk) ⊗ η(σ αk) k ! X ∗ ∗ = − tk ⊗ ψ(σ sk) ⊗ η(σ αk) (v) k = − tf(φ)(v) = tφ(v).

Thus tφ = (ψ ⊗ η) ◦ (σ∗φ) ◦ ψ−1.

In particular, over a ramiﬁcation point p ∈ R, we have

t −1 φp = ψp · φp · ψp . (3.6)

Lemma 3.2.20. Let Jr be the r × r matrix 0 I r/2 , −Ir/2 0 where Ir/2 is the identity matrix of size r/2. Let A be the set of matrices A such that t −1 A = JrAJr . Then the characteristic polynomial on A that sends A to χ(A) is a square of a polynomial in the coeﬃcients of A. In particular det(A) is a square too.

Proof. Let B = A − λIr, then

t t −1 −1 B = A − λIr = Jr(A − λIr)Jr = JrBJr , t it follows that (JrB) = −JrB, hence JrB is anti-symmetric matrix, thus

2 χ(A) = det(B) = det(JrB) = pf(JrB) , where, for an anti-symmetric matrix M, we denote by pf(M) its Pfaﬃan.

σ,+ ∼ For s ∈ W and p ∈ X, ﬁx an isomorphism (KX )p = C and let r r−1 P (x, p) = x + s1(p)x + ··· + sr(p) ∈ C[x]. Deﬁne W σ,− = {s ∈ W σ,+ | P (x, p) is square for all p ∈ R} ⊂ W σ,+. 3.2. The Hitchin system for anti-invariant vector bundles 59

Proposition 3.2.21. The Hitchin morphism induces a map

∗ σ,− σ,− T UX (r) −→ W .

σ,− Moreover, for each s ∈ W the associated spectral curve X˜s is singular and we have −1 σ,− R˜ = q (R) ⊂ S˜ = Ram(X˜s/X). And for general s ∈ W , the singular locus of X˜s is exactly R˜ and all singular points are nodes.

Proof. The first part follows directly from equation (3.6) and Lemma 3.2.20. Using Lemma 1.2.2 we deduce that the fixed locus of an involution on smooth curve is smooth, hence reduced (if it is not empty). This implies that for any s ∈ W σ,−, the associated spectral curve X˜s is singular at every point of R˜. To see that these are the only singularities for general s ∈ W σ,−, it is sufficient to show that the set of spectral data with such property is not empty in W σ,−. For this, just take the spectral data ∗ σ,− 0 r r−1 s = (0, ··· , 0, π sr) ∈ W , where sr is a general section in H (Y,KY ⊗∆ ). Moreover, for general spectral data s ∈ W σ,− the divisor R˜ has degree rn with just double points. Hence all singularity are nodes. To see that R˜ ⊂ S˜, recall that by [BNR89] Remark 3.3, the discriminant of the polynomial r ∗ r−1 ∗ x + q s1x + ··· + q sr gives the ramification divisor S˜ = Ram(X˜s/X). In other words, a point a ∈ X˜s (over p ∈ X) is in S˜ if and only if a is a multiple root of P (x, p). Hence we deduce R˜ ⊂ S˜.

It is clear that W σ,− is not a linear subspace of W σ,+. So this system is not algebraically integrable in the sense of Hitchin [Hit87].

Moreover, for general s ∈ W σ,−, over each p ∈ R, the polynomial P (x, p) is a square of a polynomial with simple roots. Thus the singularities are ordinary double points. The condition that the polynomial P (x, p) is a square of a polynomial with simple roots is given by r/2 equations, hence it decreases the dimension of W σ,+ by r/2, for each p ∈ R. More precisely, if D = C[x]6r is the vector space of polynomials of degree at most r, and S ⊂ D σ,− L is the locus of square polynomials. Then W can be deﬁned as the pullback of p∈R S via the map M W σ,+ −→ D, p∈R σ,+ which sends s ∈ W to the polynomials (P (x, p))p∈R. Since this map is a surjective linear map and because codimD(S) = r/2 we deduce r dim(W σ,−) = dim(W σ,+) − 2n 2 r2 nr = (g − 1) + − nr 2 X 2 r2 nr = (g − 1) − 2 X 2 σ,− = dim(UX (r)).

σ,− Let s ∈ W be a general spectral data such that the singular locus of X˜s is precisely R˜ and all singularities are nodes. Denote by q : Xˆs → X˜s its normalization, then the genus 60 Chapter 3. Hitchin systems g of Xˆ is given by Xˆs s

g = (arithmetic genus of X˜ ) − (number of singular points) Xˆs s 1 = r2(g − 1) + 1 − deg(R˜) (3.7) X 2 2 = r (gX − 1) + 1 − rn.

Lemma 3.2.22. Let σˆ be the lifting of the involution σ˜ to Xˆs. Then σˆ has no ﬁxed points (σˆ interchanges the two points over each singular point). Moreover, we have ∼ Xˆs/σˆ = X˜s/σ˜ = Y˜s.

Proof. If t is local parameter near p ∈ R and x is a local parameter induced by the tautological section x of the pullback of KX to |KX | in a neighborhood of a ramiﬁcation point λ ∈ R˜ over p, then by deﬁnition,σ ˜ sends t → −t and x → x, and we can write the equation of X˜s near λ as x2 + t2 + (higher terms). Then it is clear thatσ ˜ interchanges the two tangent lines at this singular point. Thus it interchanges the two branches of X˜s over λ. x

X˜ • t s λ σ˜

X • p

Now q induces a map Xˆs/σˆ −→ X˜s/σ˜ which is an isomorphism outside the branch points of X˜s → Y˜s. But we see that it is a one-to-one also over this locus. Since X˜s/σ˜ is smooth (this can be seen locally using the equation of X˜s), we deduce that this bijection is an isomorphism.

Letπ ˆ : Xˆs → Y˜s, and P = Prym(Xˆs/Y˜s), then we have

dim(P) = g − 1 (ˆπ is ´etale) Y˜s 1 = (g ˆ − 1) (Riemann-Roch) 2 Xs r2 rn = (g − 1) − (by formula (3.7)) 2 X 2 σ,− = dim(UX ).

Recall that we denoted byq ˜ : Y˜ → Y , S = Ram(Y˜ /Y ). Let ∆ˆ = det(ˆπ O )−1 and s s ∗ Xˆs Sˆ = Ram(Xˆs/X). The line bundle O(Sˆ) descends to Y˜s, and we have

O(Sˆ) = q∗O(S˜ − R˜) =π ˆ∗(O(S) ⊗ q˜∗∆−1), 3.2. The Hitchin system for anti-invariant vector bundles 61 this induces a linearization on O(Sˆ), which we call positive, and we ﬁx it hereafter. The line bundles L on Xˆs such that

σˆ∗L =∼ L−1(Sˆ), with aσ ˆ−alternating isomorphism (see Lemma 2.1.3) are those with norm (with respect toπ ˆ) equals O(S) ⊗ (q0∗∆−1) ⊗ ∆.ˆ We denote this subvariety of line bundles by Pˆ. Denote byq ˆ the map Xˆs → X. We have Theorem 3.2.23. Suppose that X → Y is ramiﬁed. For general s ∈ W σ,−, the pushforward map ˆ σ,− qˆ∗ : P 99K UX (r) σ,− is dominant. In particular UX (r) has two irreducible components. Proof. As in Theorem 3.2.12 we deduce that this map is well-deﬁned. Moreover, using Theorem 3.2.8, we deduce that the map:

∗ σ,− σ,− σ,− Π: T UX (r) → UX (r) × W is dominant. Hence, for general s ∈ W σ,−, we get a dominant map

−1 σ,− H (s) −→ UX (r).

We claim that H −1(s) is a non-empty open set of Pˆ. By Proposition 3.1.4, H −1(s) is in bijection with an open set of isomorphism classes of rank one torsion-free O −modules. X˜s Given such a torsion-free O −module , we haveσ ˜∗ =∼ ∗(S˜) (follows from the exact X˜s F F F σ,− 1 ˜ sequence (3.3)). For general s ∈ W , the divisor 2 R is reduced, consider the line bundle ∗ 1 ˜ ˆ L = q (F (− 2 R)) on Xs, it veriﬁes 1 σˆ∗L =∼ q∗(˜σ∗F (− R˜)) 2 1 =∼ q∗(F ∗(− R˜ + S˜)) 2 1 =∼ (q∗F (− R˜))−1(Sˆ) 2 =∼ L−1(Sˆ).

In fact the isomorphismσ ˆ∗L =∼ L−1(Sˆ) is induced by ψ, hence it isσ ˆ−alternating, thus L ∈ Pˆ. Conversely, given L ∈ Pˆ such thatq ˆ∗L is stable, then by duality of finite flat morphisms we deduce thatq ˆ∗L is σ−alternating anti-invariant vector bundle. We have seen that the involutionσ ˆ has no fixed point, hence Pˆ has two connected components distinguished by the parity of

0 ∗ 0 h (L ⊗ qˆ κ) = h (ˆq∗L ⊗ κ), where κ is a theta characteristic over X. It follows that the image of the two connected components of Pˆ can’t intersect. Moreover, for some σ−invariant square root α of O(R), we have κ = α ⊗ π∗κ0, where κ0 is a theta characteristic over Y . Since the σ−bilinear form ˜ ∗ ψ : E ⊗ σ E −→ OX is σ−alternating, that’s to say

ψ˜(s ⊗ σ∗t) = −ν(σ∗(ψ˜(t ⊗ σ∗s))), 62 Chapter 3. Hitchin systems

∗ where ν : σ OX → OX is the positive linearization. Taking the tensor product with α we get a bilinear form ∗ (E ⊗ α) ⊗ σ (E ⊗ α) −→ OX (R), which induces a symmetric non-degenerate bilinear form

π∗(E ⊗ α) ⊗ π∗(E ⊗ α) −→ OY .

σ,− Hence, using the result of Mumford [Mum71], the map UX (r) → Z/2 given by

0 0 E −→ h (π∗(E ⊗ α) ⊗ κ ) mod 2

σ,− is constant under deformation of E. This implies that UX (r) has two connected components.

The étalecase The étalecase is a little special. Assume that π : X → Y is étale. Recall that ∆ denotes the 2−torsion line bundle associated to π. Then we have the following results

Theorem 3.2.24. Choosing a line bundle on X of norm ∆ induces by tensor product an isomorphism σ,+ ∼ σ,− UX (r) −→ UX (r), and we have W σ,+ = W σ,−. Furthermore Π induces dominant maps

∗ σ,+ σ,+ σ,+ T UX (r) −→ UX (r) × W ,

∗ σ,− σ,− σ,− T UX (r) −→ UX (r) × W . σ,+ Moreover, for general s ∈ W , the pushforward map q∗ induces dominant rational maps

+ σ,+ − σ,− P 99K UX (r), P 99K UX (r),

+ − where P , P are diﬀerent translates of the Prym variety of X˜s → X˜s/σ˜. σ,+ σ,− In particular both UX (r) and UX (r) have two connected components.

Proof. It is clear that W σ,− = W σ,+, and for general s ∈ W σ,+, the associated spectral curve X˜s is smooth and the attached involutionσ ˜ has no ﬁxed points. Deﬁne

− −1 ˜ m ˜ P = Nmg (O(S) ⊗ ∆) ⊂ Pic (Xs).

Since we ﬁxed the positive linearization on O(S˜), it follows that the isomorphismσ ˜∗L −→∼ L−1(S˜) isσ ˜−alternating for all L ∈ P−. Moreover, let ξ be a line bundle over X of norm σ,+ ∗ −1 ∗ ∼ ∆, and E ∈ UX (r). As σ ξ → ξ is σ−alternating, then the isomorphism σ (E ⊗ ξ) −→ (E ⊗ ξ)∗ is σ−alternating too. Since q∗∆ = ∆,˜ we have

Nm(g q∗ξ) = q∗Nm(ξ) = ∆˜ , it follows that q∗ξ induces by tensor product an isomorphism P+ =∼ P−. This with Theorem 3.2.13 ends the proof of Theorem 3.2.24.

σ,− Remark 3.2.25. In this case the determinant induces a morphism det : UX (r) → P = −1 σ,− 0 −1 Nm (OY ) if r is even, and det : UX (r) → P := Nm (∆) if r is odd. 3.2. The Hitchin system for anti-invariant vector bundles 63

Trivial determinant case Here we study the case of trivial determinant σ−alternating vector bundles, denoted σ,− SU X (r). The ´etale case can be easily deduced by what has been established so far. More precisely by Theorem 3.2.24 and Remark 3.2.25 we deduce the following

Proposition 3.2.26. Assume that X → Y is ´etale,then we have

σ,− 1. If r is odd, then SU X (r) = ∅. σ,− ∼ σ,+ 2. If r is even, then SU X (r) = SU X (r). From now on we assume that π : X → Y is ramiﬁed and r is even. We use the notations of subsection 3.2.2. In particular we have the following diagram

qˆ

q q $ Xˆs / X˜s / X πˆ π˜ π q˜ Y˜s / Y.

σ,− ∗ 0 0 Lemma 3.2.27. For general s ∈ W , the pullback map qˆ : Pic (X) −→ Pic (Xˆs) is injective.

σ,− Proof. Note that for general s ∈ W , the cover X˜s → X does not factorize through an ´etalecover. So the same is true for Xˆs → X. Now apply [BL04] Proposition 11.4.3.

In particular we deduce that the Prym variety ofq ˆ : Xˆs → X is connected. ˆ ˜ Denote by Q and Q the Prym varieties of Xs → X and Ys → Y respectively, and by Qb ˆ ˜ and Qb their dual abelian varieties. Let P be the Prym variety ofπ ˆ : Xs → Ys. Proposition 3.2.28. For general s ∈ W σ,−, the intersection P ∩ Q has 22n−1 connected components.

∗ 0 0 Proof. Let µ : Q → Q be the morphism induced byπ ˜ : Pic (Y˜s) → Pic (Xˆs), and ϑ : Q → Q the one induced by Nm : Pic0(Xˆ ) → Pic0(Y˜ ). As in the proof of Xˆs/Y˜s s s Proposition 3.2.16, we deduce a commutative diagram

ϕQ Q / Qb O O µ ϑˆ

ϕQ Q / Qb .

This implies that ˆ ker(ϕQ ◦ µ) = ker(ϑ ◦ ϕQ).

∗ Since Xˆs → Y˜s is ´etale, µ has degree 2. Moreover ker(ϕQ) =q ˆ (JX [r]), hence ker(ϕQ ◦µ) = ∗ −1 ∗ 0 ∗ (ˆπ ) (ˆq (JX [r])). Now sinceπ ˆ is ´etale,for a line bundle L ∈ Pic (X),q ˆ L descends to ∗ ∗ Y˜s if and only if it isσ ˆ−invariant, and sinceσ ˆ is the lifting of σ andq ˆ is injective,q ˆ L is σˆ−invariant if and only if L is σ−invariant. So we get

∗ −1 ∗ σ ker(ϕQ ◦ µ) = (ˆπ ) (ˆq (JX [r]) ). 64 Chapter 3. Hitchin systems

σ 2n−2 Now, the locus (JX [r]) has 2 isomorphic subloci which are parameterized by the types ∗ of σ−invariant line bundles over X and the identity sublocus is π (JY [r]) (see Chapter 1 for σ more details about σ−invariant bundles and their types). It follows that card((JX [r]) ) = r2gY 22n−2. Sinceπ ˆ∗ has degree 2, we deduce that

2gY 2n−1 card(ker(ϕQ ◦ µ)) = r 2 .

2gY ˆ 2n−1 On the other hand, deg(ϕQ) = r . It follows that deg(ϑ) = 2 . This implies that ker(ϑ) has 22n−1 connected components. But by deﬁnition ker(ϑ) = P ∩ Q.

Recall that we have denoted by Sˆ = Ram(Xˆs/X) and by Pˆ the variety of line bundles L on Xˆs such that σˆ∗L =∼ L−1(Sˆ). −1 ˆ ˆ −1 Let Qˆ = Nm (δ), where δ = det(ˆq∗O ˆ ) . Xˆs/X Xs Theorem 3.2.29. For general s ∈ W σ,−, the pushforward map ˆ ˆ σ,− qˆ∗ : P ∩ Q −→ SU X (r) σ,− 2n−1 is dominant. In particular we deduce that SU X (r) has 2 connected components. Proof. Let P be the Prym variety of X → Y . It is clear that the map σ,− σ,− SU X (r) × P −→ UX (r) σ,− ˆ is surjective. By Theorem 3.2.23 we have that for general E ∈ UX (r) there exists L ∈ P th −1 such thatq ˆ∗L = E. Let λ ∈ P be an r root of det(E) ∈ P. It follows by the ∗ σ,− ∗ ˆ projection formula thatq ˆ∗(L ⊗ qˆ λ) = E ⊗ λ ∈ SU X (r). Note that L ⊗ qˆ λ ∈ P because Nm (ˆq∗λ) = O . Hence a general E ∈ SU σ,−(r) can be written as a direct image of Xˆs/Y˜s Y˜s X some L ∈ Pˆ. But since det(ˆq L) = δˆ−1 ⊗ Nm (L), ∗ Xˆs/X we deduce that ifq ˆ L has trivial determinant then Nm (L) = δˆ, thus L ∈ Qˆ. So we ∗ Xˆs/X get a dominant rational map ˆ ˆ σ,− P ∩ Q −→ SU X (r). Now we associate to any stable σ−alternating anti-invariant vector bundle E a topological invariant as follows: let ψ : σ∗E → E∗ be a σ−alternating isomorphism with determinant 1 (since E is stable, there exist only r such isomorphisms). Then over any ramification point p ∈ R take the Pfaffian pf(ψp) of the restriction of ψ to the fiber Ep. Note that pf(ψp) equals ±1. Then the type of E is defined as

τ = (pf(ψp))p∈R mod ± 1. It is clear that this is well-defined, i.e. it does not depend on the choice of ψ. Moreover the finite group P[r] of r−torsion elements of the Prym variety of X → Y acts equivariantly σ,− ˆ ˆ σ,− on SU X (r) and P ∩ Q. Actually, the action of P[r] on SU X (r) induces a free action of ∗ P[r]/(π JY [r]) on the set of types {τ} given by ¯ λ · τ = (υp · τp)p∈R mod ± 1, ¯ ∗ r/2 where λ ∈ P[r]/(π JY [r]) and υ = (υp)p is the type of the σ−invariant line bundle λ ∈ P[2] (see Chapter 1 for the definition of these types). The number of types υ is 22n−2. Since σ,− σ,− σ,− the map SU X (r) × P −→ UX (r) is surjective, P is connected and the space UX (r) has σ,− 2n−1 two connected components (Theorem 3.2.23), we deduce that SU X (r) has 2 connected components. 3.3. The Hitchin system for invariant vector bundles 65

3.3 The Hitchin system for invariant vector bundles

We have seen in Remark 1.4.5 that σ−invariant vector bundles of fixed type τ correspond to parabolic vector bundles over Y with parabolic structures associated to τ at the ramification points. The Hitchin system for parabolic vector bundles has been studied in the smooth case by Logares and Martens [LM]. In this special case we have an explicit description of the fibers of the Hitchin map depending on the considered type, as well as a dominance result as in the case of anti-invariant vector bundles. We treat also the singular case. We use results and notations of the previous section. We always suppose that the cover X → Y is ramified, the étalecase is trivial. Fix the positive linearization on OX and the negative linearization on KX (it will be clear after why we made this choice). Denote by

∗ ρ : σ KX → KX

−1 this linearization. We haveσ ¯(x) = −x, whereσ ¯ is the involution on S = P(OX ⊕ KX ) induced by these linearizations and x is the tautological section of the pullback of KX to S. Let r σ,m M 0 i W = H (KX )+. i=1 The m in the notation refers to maximal types. For simplicity, we assume in the sequel that the degree d of the σ−invariant vector bundles is 0.

σ,m Lemma 3.3.1. Let s ∈ W , X˜s the associated spectral curve, then σ lifts to an involution σ˜ on X˜s. Moreover, for general such s, we have • If r is even, then this involution has no ﬁxed point.

• If r is odd, this involution has just 2n ﬁxed points.

σ,m i Proof. Consider s = (s1, ··· , sr) ∈ W , so we have si(σ(p)) = (−1) si(p) for each point ˜ −1 p ∈ X. Let p ∈ X, and x0 = [x0 : 1] ∈ (Xs)p, then the involutionσ ¯ on P(OX ⊕ KX ) −1 ˜ attached to the ﬁxed linearizations on OX and KX sends x0 to y0 = [−x0 : 1] in (Xs)σ(p), but y0 is a solution to

r r−1 x + s1(σ(p))x + ··· + sr(σ(p)) = 0.

Indeed

r r−1 r r−1 r y0 + s1(σ(p))y0 + ··· + sr(σ(p)) = (−x0) + (−s1(p))(−x0) + ··· + (−1) sr(p) r r r−1 = (−1) x0 + s1(p)x0 + ··· + sr(p) = 0.

˜ ˜ Henceσ ¯(x0) is in (Xs)σ(p), thusσ ¯ induces an involution on Xs which we are looking for. Note that 0 is the only fixed point ofσ ˜ over a ramification point. One remarks that for odd i, si(p) = 0, for any p ∈ R. Suppose that r is odd, this implies that 0 is always in (X˜s)p, and for general s ∈ W σ,m, it is a simple root of the equation above, hence there are just 2n fixed points in X˜s. If r is even, we deduce that for general s,σ ˜ has no fixed point. 66 Chapter 3. Hitchin systems

Denote as beforeπ ˜ : X˜s → Y˜s := X˜s/σ˜. Using Riemann-Roch formula, we get for general s ∈ W σ,m: 1 1 k g ˜ = g ˜ + − , Ys 2 Xs 2 2 where k is the half of the number of ﬁxed points ofσ ˜. Let Picm(X˜ )σ˜ the locus ofσ ˜−invariant line bundles of degree m over X˜ . Since g = s s X˜s 2 r (gX − 1) + 1 and gX − 1 = 2(gY − 1) + n, we deduce from Lemma 3.3.1 ( 2 r2(g − 1) + n r + 1 r ≡ 0 mod 2 m ˜ σ˜ Y 2 dim(Pic (X ) ) = g = 2 s Y˜s 2 r −1 r (gY − 1) + n 2 + 1 r ≡ 1 mod 2 σ = dim(UX (r, 0)), where the last equality is due to Proposition 1.3.1. σ,m Remark 3.3.2. For general s ∈ W , X˜s is smooth. Indeed, taking s = (0, ··· , 0, sr) ∈ σ,m 0 r W , where sr ∈ H (KX )+ is a general global section which vanishes at most with multiplicity one at every ramiﬁcation point. Then, by the proof of Lemma 3.1.1, we deduce σ,m σ,m that X˜s is smooth. Since W is irreducible, it follows that the set of s ∈ W such that σ,m X˜s is smooth, is dense W .

Let E be a stable σ−invariant vector bundle. Recall from subsection 1.3 that we have 0 ∗ 1 ∗ considered the involutions on H (X,E ⊗ E ⊗ KX ) and H (X,E ⊗ E ) induced by the ∗ ∗ ∗ canonical isomorphism σ (E ⊗ E ) → E ⊗ E and the linearization ρ on KX (note that this is independent of the choice of a linearization on E). By Lemma 3.2.3, Serre duality is equivariant with respect to these involutions, i.e. 1 ∗ ∗ ∼ 0 ∗ H (X,E ⊗ E )+ −→ H (X,E ⊗ E ⊗ KX )+. Further we have Proposition 3.3.3. The Hitchin morphism induces a map r 0 ∗ σ,m M 0 i HE : H (X,E ⊗ E ⊗ KX )+ −→ W = H (X,KX )+. i=1 Moreover, we have an equality of dimensions σ σ,m dim(UX (r, 0)) = dim(W ). Proof. By the proof of Proposition 3.2.4, we deduce that ⊗i ∗ Hi(f(φ)) = ρ (σ (Hi(φ))), 0 ∗ where f is the involution on H (X,E ⊗ E ⊗ KX ) described above. Here, one should make a similar explicit local description of Hi, this implies the ﬁrst part of the lemma. As we use the negative linearization on KX , by Remark 1.1.3, it follows  h0(X,Ki ) = h0(Y,Ki ⊗ ∆i) = (2i − 1)(g − 1) + in i ≡ 0 mod 2  X + Y Y 0 i 0 i i−1 h (X,KX )+ = h (Y,KY ⊗ ∆ ) = (2i − 1)(gY − 1) + (i − 1)n i ≡ 1 mod 2, i > 3 .  0 0 h (X,KX )+ = h (Y,KY ) = gY Taking the sum, we get

r ( 2 r2 X r (gY − 1) + n + 1 r ≡ 0 mod 2 h0(X,Ki ) = 2 . X + 2 r2−1 i=1 r (gY − 1) + n 2 + 1 r ≡ 1 mod 2 3.3. The Hitchin system for invariant vector bundles 67

3.3.1 Smooth case Recall from Remark 1.3.2 that we have deﬁned a maximal type to be a type τ such that σ,τ UX (r, 0) has maximal dimension, and we have denoted the set of such types by MAX. σ,m Theorem 3.3.4. Let s ∈ W such that X˜s is smooth. Then the direct image map induces a dominant map m ˜ σ˜ σ,m q∗ : Pic (Xs) 99K UX (r, 0), σ,m where UX (r, 0) is the moduli space of σ−invariant vector bundles of type τ ∈ MAX (see Remark 1.3.2). Moreover, for each type τ ∈ MAX, there exists a unique types τ˜ of invariant line bundles in Pic(X˜s), such that we have a dominant map m ˜ σ,˜ τ˜ σ,τ q∗ : Pic (Xs) 99K UX (r, 0).

Proof. It is clear that L isσ ˜−invariant if and only if q∗L is σ−invariant. By the proof of Theorem 3.2.12, for general σ−invariant vector bundle E of type τ ∈ MAX, the restriction of the Hitchin map 0 ∗ σ,m HE : H (X,E ⊗ E ⊗ KX )+ → W is dominant. This implies that the map ∗ σ,m σ,m σ,m Π: T UX (r, 0) −→ UX (r, 0) × W is dominant too. Moreover, if we fix a type τ ∈ MAX, then there exists a corresponding typeτ ˜ ofσ ˜−invariant line bundles on X˜s, such that m ˜ σ,˜ τ˜ σ,τ Pic (Xs) 99K UX (r, 0) is dominant. The typeτ ˜ is constructed as follows: Suppose that r is odd (the even case is trivial sinceσ ˜ has no fixed point by Lemma 3.3.1). Remark first thatτ ˜ ∈ {+1, −1}2n/±. if p ∈ R is such that kp = (r + 1)/2, then over such p, take −1 inτ ˜, and +1 over the rest of points in R (note that, because s ∈ W σ,m is general, over each p ∈ R there is just one fixed point byσ ˜, so we identify R˜ = Fix(˜σ) with R in this case). If L isσ ˜−invariant line bundle ˜ ∼ L over Xs of typeτ ˜, then using the identification (q∗L)p = x∈q−1(p) Lx, we see easily that the type of q∗L is τ.

We prove in the next section that the only types corresponding to smooth spectral covers of X are the maximal ones.

3.3.2 General case Now let τ be any type, for simplicity of notations we suppose that we have just one r point p ∈ R such that k < [ ], where [ ] stands for the floor function. In fact we can p 2 always suppose kp 6 [r/2] due to taking the tensor product by OX (p). And for all other ramification point a 6= p, we assume that ka is maximal (that’s ka = r/2 if r is even, and ka = (r ± 1)/2 if r is odd). To get a vector bundle of such type as a direct image, we should have r − 2kp fixed points byσ ˜ above p. Indeed, let x be a point of X˜s above p, asσ ˜ interchanges the two fibers Lx and Lσ˜(x), its matrix over these two points is given by 0 1 1 0 ∼ . 1 0 0 −1

Hence, one should have 2kp non-ﬁxed points to get the −1 eigenvalue with multiplicity kp. This intuition is proved in the following theorem 68 Chapter 3. Hitchin systems

Theorem 3.3.5. Let τ be a type as above, and denote by r σ,τ 0 0 2kp+1 M 0 i W = H (KX )+ ⊕ · · · ⊕ H (KX )+ ⊕ H (X,KX (−(i − 2kp − 1)p))+. i=2kp+2 Then, for any σ−invariant stable vector bundle E of type τ, the Hitchin map factorizes through W σ,τ given a map

0 ∗ σ,τ HE : H (E ⊗ E ⊗ KX )+ −→ W . Moreover, σ,τ σ,τ dim(UX (r, 0)) = dim(W ). Proof. First we verify the dimensions. By Proposition 1.3.1, we have

( r2 σ,τ 2 (2n − 1) 4 if r ≡ 0 mod 2 dim(UX (r, 0)) = r (gY − 1) + 1 + kp(r − kp) + r2−1 . (2n − 1) 4 if r ≡ 1 mod 2

Recall that we have ﬁxed the negative linearization on KX . By Lefschetz ﬁxed point formula we deduce that  kn − i k ≡ 0, i ≡ 0 mod 2  2  i+1 0 k kn − 2 k ≡ 0, i ≡ 1 mod 2 h (X,K (−ip))+ = (2k − 1)(gY − 1) + . (3.8) X (k − 1)n − i k ≡ 1, i ≡ 0 mod 2  2  i−1 (k − 1)n − 2 k ≡ 1, i ≡ 1 mod 2 So the dimension of W σ,τ is given by

r−2kp−1 X dim(W σ,τ ) = dim(W σ,m) − d(i), i=1 where ( i 2 if i ≡ 0 mod 2 d(i) = i+1 . 2 if i ≡ 1 mod 2 By a simple computation, we get

r−2kp−1 ( r2 X 4 − kp(r − kp) if i ≡ 0 mod 2 d(i) = r2−1 . i=1 4 − kp(r − kp) if i ≡ 1 mod 2 It follows that  2 2 r r (gY − 1) + 1 + (2n − 1) + kp(r − kp) if r ≡ 0 mod 2 dim(W σ,τ ) = 4 r2 − 1 r2(g − 1) + 1 + (2n − 1) + k (r − k ) if r ≡ 1 mod 2 Y 4 p p σ,τ = dim(UX (r, 0)).

0 ∗ ∗ ∼ Now take φ ∈ H (X,E ⊗ E ⊗ KX )+ and let ϕ : σ E −→ E be a linearization. Then the following diagram commutes

φ : E / E ⊗ K O X ϕ σ∗(ϕ⊗ρ)

∗ ∗ ∗ ∗ σ φ : σ E / σ E ⊗ σ KX , 3.3. The Hitchin system for invariant vector bundles 69 that’s σ∗φ = σ∗(ϕ ⊗ ρ) ◦ φ ◦ ϕ. In particular over p, one has φp = −ApφpAp, where Ap = diag(−1, ··· , −1, 1, ··· , 1) is r × r diagonal matrix. | {z } kp This implies that the matrix φp is of the form 0 M , N 0 where M and N are two matrices of type kp × (r − kp) and (r − kp) × kp.

0 Now let t be a local parameter on the neighborhood of p, and denote φ(t) = φp + tφ the restriction of φ to this neighborhood. Then we get

0 sr(t) = det(φ(t)) = det(φp + tφ ). But t × ∗ M 0 φ + tφ0 = , (3.9) p N 0 t × ∗ 0 0 where M and N are two matrices of type kp × (r − kp) and (r − kp) × kp respectively, which are not necessarily divisible by t. Using the development of the determinant as a sum of monomials in the entries of the matrix, we see that every monomial contains at 0 0 least r − 2kp entries that belong to neither M nor N , i.e. divisible by t. So we deduce that det(φ(t)) is divisible by tr−2kp , hence

0 r det(φ(t)) ∈ H (X,KX (−(r − 2kp)p))+. But we have

0 r 0 r H (X,KX (−(r − 2kp − 1)p))+ = H (X,KX (−(r − 2kp)p))+, because the ﬁrst space is included in the second one and they have the same dimension by formula (3.8).

The general case is treated similarly. Let i > 2kp + 2. Consider φ(t) as an element of i i Matr(C[[t]]), and denote it by φ(t) = (ai,j(t))i,j. As Hi(φ) is by deﬁnition (−1) Tr(Λ φ(t)). We just need to calculate the diagonal elements of the matrix Λiφ(t). Assume that

i Λ φ(t) = (αk,l(t)), where k and l are i-tuples of strictly increasing integer in {1, ··· , r}. Then if k = (k1 < ··· < ki) we have α (t) = det(a (t)) 0 . k,k kl,kl0 16l,l 6i

Hence, from the form of φ(t) given in (3.9), we deduce that Hi(φ(t)) is divisible by at least ti−2kp , hence

0 i 0 i Hi(φ) ∈ H (X,KX (−(i − 2kp)p))+ = H (X,KX (−(i − 2kp − 1)p))+. Thus σ,τ HE(φ) ∈ W . 70 Chapter 3. Hitchin systems

Unfortunately, most of the types correspond to singular spectral curve X˜s → X. But σ,τ for general s ∈ W , where τ is as above, the corresponding X˜s has just the point 0 over p which is singular with multiplicity r − 2kp. Moreover, this singularity is ordinary (the tangents at this point are distinct), we can see that using the equation deﬁning X˜s and the σ,τ generality of s ∈ W . Hence the geometric genus of the normalization Xˆs of X˜s is equals to 2 (r − 2kp)(r − 2kp − 1) g ˆ = r (gX − 1) + 1 − . Xs 2

Moreover, the involutionσ ˜ lifts to an involutionσ ˆ on Xˆs with r − 2kp fixed points if r is even and r−2kp +2n−1 if r is odd (Recall that we assumed for simplicity that we have just one point p ∈ R with kp < [r/2]). Indeed, if t is a local parameter in a local neighborhood of p and x is a local parameter near the ramification point 0 ∈ R˜ over p, then by definition, σ˜ sends t → −t and x → −x, thus it does not interchange the two branches near λ x

σ˜ X˜s • t λ

X • p

If r is odd,σ ˆ ﬁxes also the ﬁxed points ofσ ˜ outside the singular points.

Let ε(r) = r mod 2. It follows that the genus g of Yˆ = Xˆ /σˆ is given by Yˆs s s 1 r − 2kp 2n − 1 g ˆ = g ˆ + 1 − − ε(r) Ys 2 Xs 2 2 r2 (r − 2k )2 2n − 1 = (g − 1) + 1 − p + ε(r) 2 X 4 4 r2 (r − 2k )2 2n − 1 = r2(g − 1) + 1 + n − p + ε(r) Y 2 4 4 r2 − ε(r) = r2(g − 1) + 1 + (2n − 1) + k (r − k ). Y 4 p p This implies g = dim(Picm(Xˆ )σˆ) = dim(U σ,τ (r, 0)). Yˆs s X So the generic ﬁber of the Hitchin morphism

∗ σ,τ σ,τ H : T UX (r, 0) −→ W is of maximal dimension. So we get again the complete integrability of the Hitchin system in this case too. Moreover, with the same method used so far proving the dominance results, we deduce Theorem 3.3.6. For each type τ, the pushforward map m ˆ σ,ˆ τˆ σ,τ Pic (Xs) 99K UX (r, 0) is dominant, for some type τˆ of σˆ−invariant line bundles over Xˆs. 3.3. The Hitchin system for invariant vector bundles 71

Proof. First the typeτ ˆ is constructed as follows: If r is even,σ ˆ has r − 2kp fixed points which are all over p, then since kp is chosen strictly smaller than [r/2], we take the trivial type over these r − 2kp fixed points. If r is odd, then over any ramification point a such r+1 that ka = 2 the type is equal −1, and over the rest of ramification points other than p the type is +1. Now we deduce as in the proof of Theorem 3.2.12 that the map

∗ σ,τ σ,τ σ,τ Π: T UX (r, 0) −→ UX (r, 0) × W

σ,τ −1 σ,τ is dominant. So for general s ∈ W , the ﬁber H (s) dominates UX (r, 0). Moreover H −1(s) is identiﬁed, by Proposition 3.1.4, with a set of torsion-free rank one sheaves over X˜s, which areσ ˜−invariant. Let x0 be the singular point of X˜s over p. Then twisting these torsion-free sheaves with

(r − 2kp)(r − 2kp − 1) O ˜ (− x0) Xs 2

−1 m σ,ˆ τˆ and pulling them back to Xˆs, we identify H (s) with the open subset of Pic (Xˆs) of line bundles such thatq ˆ L is stable, where m is the degree of (ˆq O )∗ (m = r(r − 1)(g − ∗ ∗ Xˆs X (r−2kp)(r−2kp−1) 1) − 2 ). The result follows.

Chapter 4

Conformal Blocks

σ,+ σ,τ Recall that we have denoted by SU X (r) (resp. SU X (r)) the moduli stack of σ−symmetric (resp. σ−alternating of type τ) anti-invariant vector bundles with trivial σ,− determinant. When the type τ is the trivial one, we denote by SU X (r) the corresponding moduli stack. The main topic of this chapter is the study of some line bundles over the σ,± moduli stacks SU X (r) and the spaces of their global sections, called generalized theta σ,− functions. It turns out that the restriction of the determinant bundle to SU X (r) has a square root associated to each σ−invariant theta characteristic κ over X, we denote Pκ this line bundle, and call it the Pfaffian of cohomology line bundle. However, in the ramified σ,+ case, this is not true for the stack SU X (r) of σ−symmetric bundles. In the étalecase, the two stacks are isomorphic (for even rank, see Theorem 3.2.24). In this case also, the determinant bundle admits some square roots. Our main result is the identification of the spaces of global sections of the powers of σ,− the Pfaffian line bundle Pκ (resp. the determinant line bundle D) on SU X (r) (resp. σ,+ SU X (r)) with the conformal blocks spaces Vσ,−(k) and Vσ,+(k) associated to some twisted affine Kac-Moody Lie algebras respectively.

4.1 Preliminaries on twisted Kac-Moody algebras

In this first section, we recall briefly the construction of the twisted affine Kac-Moody Lie algebras and the attached conformal blocks. We use notations of [Kac90]. The definition of twisted conformal blocks is adapted from [Hon16] (which in turn adapted from [FS04], where a more general definition is given in the framework of vertex algebras). Consider an outer automorphism τ of the Lie algebra slr(C). It is an order two automorphism. The involution τ is extended to an automorphism of the affine Kac-Moody algebra Lb(slr) = slr(K ) ⊕ CK, where K = C((t)) and K a central element, by sending x ⊗ g(t) to τ(x) ⊗ g(−t) and fixes the center. Then the fixed subalgebra of this involution, (2) denoted by Lb(slr, τ), is an affine Lie algebra of type Al (after adding a scaling element D), where l = br/2c, and it is called twisted affine Lie algebra. Let ˚g be the finite simple Lie algebra of L(slr, τ) (see [Kac90] §6.3 for a precise definition). Then ˚g is of type Cl if r τ is odd, and is isomorphic to the fixed subalgebra slr(C) if r is even. Since we will be interested mainly in the following two involutions

+ t − t −1 σ (a(t)) = − a(−t), σ (a(t)) = −Jr a(−t)Jr , where 0 Ir/2 Jr = , −Ir/2 0 ± we give an explicit constructions of L(slr, σ ). Let Mi,j be the canonical basis of the vector space of square matrices of size r. Let 0 0 ∗ h ⊂ slr(C) be the Cartan subalgebra of diagonal matrices and let α1, ··· , αr−1 ∈ h be the 74 Chapter 4. Conformal Blocks

∗ ∗ 0 0 0 0 simple roots deﬁned by αi = Mi,i − Mi+1,i+1. Denote by E1, ··· ,Er−1 and F1, ··· ,Fr−1 0 0 t 0 0 ∨ the Chevalley generators of slr(C): Ei = Mi,i+1, Fi = − Ei. Let α0 = M1,1 − Mr,r, 0 0 t 0 E0 = M1,r and F0 = − E0. Then the Chevalley generators of Lb(slr) are given by 0 −1 0 e0 = t ⊗ E0, f0 = t ⊗ F0, and for i ∈ {1, ··· , r − 1} 0 0 ei = 1 ⊗ Ei, fi = 1 ⊗ Fi .

Recall the Lie bracket on Lˆ(slr) is given by dP [g(t), h(t)] = [g, h] ⊗ P (t)Q(t) + (g, h)Res( Q)K, dt where g, h ∈ slr, P,Q ∈ K and ( , ) is the normalized Killing form on slr. Moreover, by extending the linear forms αi to h ⊕ CK such that αi(K) = 0, then αi are the simple roots of Lb(slr).

− Case L(slr, σ ). Let r = 2l. This is the algebra constructed in [Kac90] Page 128. We − 0 0 0 0 0 0 can assume, after conjugation, that σ sends Ei to Er−i, Fi → Fr−i and αi → αr−i. So let’s deﬁne ∨ 0∨ 0∨ ∨ 0∨ ∨ 0∨ 0∨ 0∨ • αi = α i + α r−i (1 6 i 6 l − 1), αl = α l and α0 = −2α 0 + α 1 + α r−1. 0 0 0 0 0 • Ei = E + E (1 i l − 1), El = E and E0 = E 0 0 − E 0 0 . i r−i 6 6 l −α0+αr−1 −α0+α1 0 0 0 0 0 • Fi = F + F (1 i l − 1), Fl = F and F0 = −E 0 0 + E 0 0 . i r−i 6 6 l α0−αr−1 α0−α1 − The Chevalley generators of Lb(slr, σ ) are given by

ei = 1 ⊗ Ei, fi = 1 ⊗ Fi for i = 1, . . . , l. −1 e0 = t ⊗ E0, f0 = t ⊗ F0. ∨ ∨ ∨ ∨ Consider the elementsα ˜i = 2αi /(αi , αi ) ∈ h. Since the normalized bilinear form ( ; ) ∼ ∗ ∨ is non-degenerate on h it induces an isomorphism h = h . So letα ˜i be the images ofα ˜i − under this bijection. Then the simple roots of Lb(slr, σ ) are given by 1 α = ⊗ α˜ , 0 2 0

αi = 1 ⊗ α˜i, i = 1, . . . , l. ∨ ∨ The simple coroots are just 1 ⊗ αi , for i = 1, . . . , l. We denote them again by αi . For i = 0 ∨ ∨ the simple coroot is 2K + 1 ⊗ α0 . We denote it also by α0 . − In particular, the normalized bilinear form on Lb(slr, σ ) is given by 1 (P ⊗ x; Q ⊗ y) = Res(t−1PQ)(x; y), 2 − where ( , ) is the normalized bilinear form on slr(C). The 2−cocycle on L(slr, σ ) that − deﬁnes Lb(slr, σ ) is given by 1 dg ψ(g(t), h(t)) = Res(Tr( h)). 2 dt

+ Case L(slr, σ ). We treat the case r = 2l (the odd case is again treated in [Kac90]). + 0 0 0 0 0 0 We can assume, after conjugation, that σ sends Ei to −Er−i, Fi → −Fr−i and αi → αr−i. So we deﬁne the following elements of sl2l : 4.1. Preliminaries on twisted Kac-Moody algebras 75

∨ 0∨ 0∨ ∨ 0∨ ∨ 0∨ 0∨ 0∨ • βi = α l−i + α l+i (1 6 i 6 l − 1), βl = α 0 and β0 = 2α l + α l−1 + α l+1. 0 0 0 0 0 • Ei = Ei − Er−i (1 i l − 1), El = E0 and E0 = E 0 0 − E 0 0 . 6 6 αl+αl+1 αl+αl−1 0 0 0 0 0 • Fi = Fi − Fr−i (1 i l − 1), Fl = F0 and F0 = −E 0 0 + E 0 0 . 6 6 −αl−αl+1 −αl−αl−1

+ Remark that the aﬃne node β0 of Lb(slr, σ ) is then the node αl with the notation of Table Aﬀ 2 of [Kac90] Page 55. Thus when deleting this node the remaining diagram is of type Dl.

+ As before, we deﬁne the Chevalley generators of Lb(slr, σ ) by

ei = 1 ⊗ Ei, fi = 1 ⊗ Fi for i = 1, . . . , l.

−1 e0 = t ⊗ E0, f0 = t ⊗ F0.

The simple coroots of the simple invariant Lie algebra (= so2l) are given by

˜∨ ∨ ∨ ∨ βi = 2βi /(βi , βi ), i = 0, . . . , l.

∗ As above denote by β˜i the corresponding elements of h . Then the simple roots of + Lb(sl2l, σ ) are given by β0 = 2K + 1 ⊗ β˜0,

βi = 1 ⊗ β˜i, i = 1, . . . , l. + From the construction of Lb(slr, σ ), it is clear that the Coxeter coeﬃcients and their dual in this case are taken in the inverse order. We recall the dual Coxeter coeﬃcients of ± the twisted Kac-Moody algebras Lb(slr, σ ) in the following table.

∨ ∨ ∨ ∨ ∨ a0 a1 a2 ··· al−1 al + Lb(sl2l, σ ) 2 2 2 ··· 1 1 − (4.1) Lb(sl2l, σ ) 1 1 2 ··· 2 2 + Lb(sl2l+1, σ ) 1 2 2 ··· 2 2 Dual Coxeter coeﬃcients.

Now, when we add a scaling elements to the above algebras, i.e. derivations D± such that n n [D±, t ⊗ x] = nt ⊗ x, ˆ ± then, by [Kac90] Theorem 8.5, both Kac-Moody algebras L(slr, σ ) ⊕ CD± are isomorphic to the Kac-Moody algebra g(A), where A is the aﬃne generalized Cartan matrix of type (2) Ar−1. In particular, we deduce an isomorphism

ˆ + ∼ ˆ − L(slr, σ ) ⊕ CD+ = L(slr, σ ) ⊕ CD−.

± Moreover, the derivations D± induces a weight decomposition of the algebras L(slr, σ ) ⊕ CD±. The main observation is that the above isomorphism does not respect the decom- ± position L(slr, σ ) ⊕ CK ⊕ CD±. + We will see in a moment that under the above isomorphism, the fundamental weight λ0 ˆ + − of L(slr, σ ) ⊕ CD is sent to twice the fundamental weight λ0 . 76 Chapter 4. Conformal Blocks

Twisted conformal blocks ± ± ± Let λ0 , ··· , λl be the fundamental wights of the twisted aﬃne Lie algebras Lb(slr, σ ), ± i.e. λi are linear forms on the Cartan subalgebras such that

+ − λi (βj) = λi (αj) = δij, i, j = 0, . . . , l.

± Denote by ˚g ⊂ L(slr, σ ) the simple Lie algebra generated by ei and fi for i = 1, ··· , l. + Note that ˚g is of type Dl in the case of σ when r is even, and it is of type Cl otherwise. Moreover, we have the identiﬁcations

± ˚ ∨ ± λi = λi + ai λ0 , i = 1, . . . , l. where ˚λi (i = 1, . . . , l) are the fundamental weights of ˚g.

+ ∨ Remark 4.1.1. Remark that, for an even rank r, the weight λ0 has level equals a0 = 2, − ∨ while λ0 has level a0 = 1 (see Table 4.1).

σ,± ± Denote by P the set of dominant integral weights of Lb(slr, σ ). By [Kac90] §12.4 (see also [Hon16] Lemma 6.1), one deduces a bijection between Pσ,± and the set

˜σ,± ˚ ± P = {(λ, k)|λ ∈ P, λ, % 6 k}, where ˚P is the set of dominant weights of ˚g, and %− is the highest coroot of ˚g when r is even, and %+ is twice the highest short coroot of ˚g when r is odd.

± σ,± For µ ∈ P , denote by Hµ± (k) the irreducible highest weight module of level k of ± ± −→± ± ± σ,± Lb(slr, σ ) of highest weight µ . Let µ = (µ1 , ··· , µ2n) be a vector of elements of P parameterized by the points of R, and deﬁne

H−→µ ± (k) = H ± (k) ⊗ · · · ⊗ H ± (k). µ1 µ2n

0 Finally, let AR = H (X r R, OX ). By considering the associated Lorrent series at p ∈ R, σ± σ± σ± we get an inclusion slr(AR) ⊂ slr(Kp) . We can than deﬁne an action of slr(AR) on H−→µ ± (k) as product of representations (i.e diagonal action). More explicitly, for α ∈ σ± slr(AR) and X = X1 ⊗ · · · ⊗ X2n, we have X α · X = X1 ⊗ · · · ⊗ α · Xi ⊗ · · · ⊗ X2n. i

−→± ± Deﬁnition 4.1.2. The conformal block attached to the data (X, σ, R, µ , Lb(slr, σ ),k) is deﬁned by h i∗ −→ Vσ,±(k) = H µ ± (k) σ± , slr(AR) where for a g−module V , we denote by Vg the space of coinvariants of V , thus the largest quotient of V on which g acts trivially.

In [Hon16], it is shown that these are ﬁnite dimensional vector spaces and a formula for their dimensions is conjectured (a Verlinde formula for twisted conformal blocks). 4.2. Loop groups and uniformization theorem 77

4.2 Loop groups and uniformization theorem

4.2.1 Uniformization theorem

For a ramification point p ∈ X, denote by Op the completion of the local ring at p, Kp its fraction field and Vp a complementary vector subspace of Op in Kp. Let SU X (r) denote the moduli stack of rank r vector bundles over X with a trivialization of its determinant. ∗ Let’s fix the canonical linearization on OX , so we identify σ OX and OX . Moreover, since all the types are isomorphic, we assume hereafter that τ = (+1, ··· , +1) mod ± 1 and σ,− denote the corresponding moduli stack by SU X (r). In [BL94], it is proved that ∼ SU X (r) = SLr(Op)\SLr(Kp)/SLr(Ap), 0 ∼ ∼ where Ap = H (X −p, OX ). Let t be a local parameter at p, then Kp = C((t)), Op = C[[t]]. ± Consider the two involutions σ on SLr(Kp) given by g(t) → σ+(g(t)) = tg(−t)−1, − t −1 −1 g(t) → σ (g(t)) = Jr · g(−t) · Jr , where 0 Ir/2 Jr = , −Ir/2 0

Ir/2 is the identity matrix of size r/2. Note that

pf(Jr) = 1.

Let Q = SLr(Op)\SLr(Kp). In [PR08b], it is proved that σ+ σ+ σ+ Q = SLr(Op) \SLr(Kp) . σ+ σ+ Note that SLr(Op) is the maximal parahoric subgroup of SLr(Kp) and, with the notations of loc. cit. this case corresponds to I = {0}. In fact, their involution is the + conjugation of σ by the anti-diagonal matrix Dr with all entries equal 1. But this does t not change much. Indeed, by taking a matrix A such that Dr = AA (such matrix can be σ+ constructed easily), then conjugation by A realizes an isomorphism between SLr(K ) and their invariant locus. σ,± σ± σ± We denote in the sequel by Q the quotient SLr(Op) \SLr(Kp) , for some p ∈ R. Theorem 4.2.1. We have an isomorphism of stacks σ,± ∼ σ± σ± σ± SU X (r) = SLr(Op) \SLr(Kp) /SLr(Ap) . σ,± σ,± Moreover, the projections Q → SU X (r) are locally trivial for the fppf topology.

Proof. Let G and Hτ the invariant Weil restrictions of the constant group scheme SLr (see Proposition 2.2.1). Using the main Theorem of [Hei10], we deduce, for a ramiﬁcation point p ∈ X over a branch point y ∈ Y , that ∼ 0 MY (G) = G(Oy)\G(Ky)/H (Y r y, G) ∼ σ+ σ+ σ+ = SLr(Op) \SLr(Kp) /SLr(Ap) . ∼ 0 MY (Hτ ) = Hτ (Oy)\Hτ (Ky)/H (Y r y, Hτ ). 0 ∼ σ− Since the type τ is assumed to be trivial, H (Y r y, Hτ ) = SLr(Ap) . Thus ∼ σ− σ− σ− MY (Hτ ) = SLr(Op) \SLr(Kp) /SLr(Ap) . Now it suﬃces to apply Proposition 2.2.2. 78 Chapter 4. Conformal Blocks

4.2.2 The Grassmannian viewpoint Note that Qσ,+ is an ind-variety, which is a direct limit of a system of projective σ,+ σ,+ 0 σ+ N σ+ N varieties (QN )N>0, the QN are the quotients (S ) \(S ) , where S is the sub- −1 scheme of SLr(K ) parameterizing matrices A(t) such that A(t) and A(t) have poles σ,τ of order at most N. As we said above, since all the stacks SU X (r) are isomorphic, so for simplicity we assume that τ is the trivial type. So let’s denote Qσ,− the quotient σ− σ− σ,− σ,− SLr(Op) \SLr(Kp) . This is again an ind-variety direct limit of (QN )N>0, the QN are the quotients (S0)σ− \(SN )σ− . 0 N By [BL94] Proposition 2.4, the varieties QN := S \S are identiﬁed with subvarieties (with the same underline topological spaces) of the Grassmannian Grt(rN, 2rN) of t−stable r −N ⊕r N ⊕r subspaces of dimension rN of FN := t O /t O . r r Consider the σ−Hermitian forms Ψ± : K × K −→ K deﬁned by

r t X Ψ+(v, w) = v · σ(w) = viσ(wi), i=1

t Ψ−(v, w) = v · Jr · σ(w), t t r σ± where v = (v1, ··· , vi) and w = (w1, ··· , wi) are in K . Then the groups SLr(K ) can be deﬁned as the loci of matrices A ∈ SLr(K ) which are unitary with respect to the r forms Ψ±, i.e. Ψ±(A · v, A · w) = Ψ±(v, w) for all v, w ∈ K . −N r r Consider the forms Ψ˜ ± on t O ⊂ K deﬁned as the composition

˜ −N ⊕r −N ⊕r Ψ± −2N Res Ψ± : t O × t O −−→ t O −−→ C, ˜ N ⊕r −N ⊕r where Res : K → C is the residue map. The forms Ψ± vanish on t O ⊂ t O , hence ˜ r they induce two forms, denoted again by Ψ±, on FN ˜ r r Ψ± : FN × FN −→ C. ˜ r ˜ Lemma 4.2.2. Ψ+ is an anti-symmetric non-degenerate bilinear form on FN , while Ψ− is a symmetric non-degenerate bilinear form.

k Proof. Consider the canonical basis of the vector space Fr given by the classes of t for r r k = −N, ··· ,N −1. It induces a canonical basis of FN . Then for v = (vi)i, w = (wi)i ∈ FN , the forms Ψ± are given explicitly in this basis by

r N−1 X X −j−1 i i Ψ+(v, w) = (−1) ajb−j−1, i=1 j=−N

r N−1 X X −j−1+ε(i) i r−i Ψ−(v, w) = (−1) ajb−j−1, i=1 j=−N i i where ε(i) equals 1 if i ∈ {1, ··· , r/2}, and 0 otherwise, and vi = (aj), wi = (bj) are in FN . From this the result follows easily. We have the following

σ,± Proposition 4.2.3. The spaces QN are isomorphic to closed subvarieties (with the same t,σ underline topological subspaces) of the isotropic Grassmannian Gr± (rN, 2rN) which pa- ˜ r rameterizes Ψ±−isotropic t−stable vector subspaces of FN of dimension rN. 4.2. Loop groups and uniformization theorem 79

Proof. We prove it for the symmetric case, the other one follows similarly. ⊕r r ˜ N σ+ The image of O in FN is Ψ+−isotropic, hence, for every A(t) ∈ (S ) , the correspond- t σ,+ ˜ ing point in Gr (rN, 2rN) of the class of A(t) in QN is Ψ+−isotropic. Thus it is in t,σ Gr+ (rN, 2rN). Conversely, assume that we have a point W of the isotropic Grassmannian. Let A(t) ∈ SN be a representative of the corresponding class in S0\SN . We have for every v, w ∈ Or, r −k Ψ+(A · v, A · w) ∈ O, to see this assume that for some v, w ∈ O , the coefficient of t k of Ψ+(A · v, A · w) is nonzero (k > 0), then one deduces that Res(Ψ+(A · (t v),A · w)) = k Ψ˜ +(A·(t v),A·w) 6= 0, contradiction. Now let (ei)i be the canonical basis of the K −vector r 0 space K and let B(t) = (Ψ+(A · ei,A · ej))i,j, we see that B(t) ∈ SLr(O) = S , and we have by definition B(t) = tA(t)A(−t). In particular we see that B(t) = tB(−t), hence t B(t) = C(−t) · C(t) for some C(t) ∈ SLr(O), and C(t)A(t) is also a representative of W and it is of course in (SN )σ+ . In other words the corresponding point of W in S0\SN is in σ,+ QN . This proves the proposition. 0 N Consider the variety QN = S \S which is as a topological space isomorphic to the t Grassmannian Gr (rN, 2rN). Fix an identification of QN as subspace of the homogeneous space SL2rN (C)/PN , where PN is the parabolic subgroup of SL2rN of matrices of the form AB , 0 C where A, B and C are square rN ×rN matrices. Let OQN (1) be the line bundle attached to ∗ −1 the character χ :P → C which sends a matrix as above to det(A ). It is well known that the Picard group of QN is infinite cyclic generated by OQN (1) (it is actually isomorphic to the character group of the maximal parabolic subgroup PN ). σ,− Proposition 4.2.4. The restriction of OQN (1) to QN has a square root, which we denote σ,− by O σ,− (1). It is in fact the generator of the Picard group of Q . QN N σ,− Proof. By Proposition 4.2.3, the variety QN is isomorphic to a subvariety of the classical 0 0 Grassmannian SO2rN (C)/PN , where PN = PN ∩SO2rN (C). The restriction of the character 0 χ to PN is denoted again by χ. Now, consider the universal cover of SO2rn(C), which is ˜ the Spin group Spin2rN (C). It is a double cover of SO2rN (C). Let PN ⊂ Spin2rN (C) the 0 ˜ inverse image of PN . Then, by [DSS92] Chapter 5, Theorem 3.3.1, the lifting of χ to PN has a square root which we denote by χ−. Since we have ˜ ∼ 0 Spin2rn(C)/P = SO2rn/PN , σ,− we deduce that the line bundle over QN attached to χ− is the square root of the restriction of OQN (1). ˜ The Picard group of Spin2rN /PN is infinite cyclic isomorphic to the character group of PÑ , which is generated by χ−. This implies the second claim.

Proposition 4.2.5. The ind-varieties Qσ,± are integral. Proof. We know already that Qσ,± are connected, hence they are irreducible. Moreover, By [PR08b] Theorem 0.2, the ﬂag varieties Qσ,± are reduced. Furthermore, we can show directly that the ﬂag varieties Qσ,± are reduced. We follow the −1 same method as in [BL04]. First denote by SLr(K )0 = SLr(C[t ])SLr(O). We know (cf. loc. cit.) that SLr(K )0 is an open and that gSLr(K )0, for g ∈ SLr(K ), cover SLr(K ). σ± σ± σ± We deduce easily that SLr(K )0 is also an open and gSLr(K )0 , for g ∈ SLr(K ) , σ± covers SLr(K ) . Now we claim the following 80 Chapter 4. Conformal Blocks

−1 σ± Claim. The ind-varieties SLr(C[t ]) are reduced. ± −1 σ± −1 Proof. Denote by VN ⊂ SLr(C[t ]) the loci of matrices of degree (in t ) at most N. −1 σ± ± ± Then SLr(C[t ]) is a direct limit of VN , hence it is sufficient to prove that VN are ± reduced. In fact VN are even smooth. N+1 −1 Let Mr be the vector space over C of square matrices of size r. Denote by Mr ⊂ Mr[t ] the subspace of polynomials of degree at most N with coefficients in Mr. One can easily + − check that VN (resp. VN ) is exactly the locus of elements of determinant 1 of the fiber N+1 2N+1 over the identity matrix of the map α± : Mr −→ Mr given by

t t −1 α+(A(t)) = A(−t) · A(t) (resp. α−(A(t)) = A(−t) · Jr · A(t) · Jr ).

Remark that the elements of these fibers have determinants in {±1}. For simplicity, we will consider just σ+ case hereafter. The image of α+ lives inside the 2N+1 τ 2N+1 vector subspace (Mr ) of τ-invariant matrices, where τ is the involution on Mr t + defined by τ(A(t)) = A(−t). It is easy to see that the fiber of α over Ir is given by r(r − 1) (2N + 1) + r(N + 1) 2

d quadratic equations. They can be calculated explicitly: Let Xi,j be the canonical dual basis N+1 of Mr , where i, j are the indices of the coeﬃcients of the matrices and d corresponds to the powers of t−1. Then the equations of the ﬁber can be written explicitly in the form

r d X X m n m fij = (−1) XikXkj − δijδd0, k=1 n+m=d where 1 6 i 6 j 6 r, n and m are between 0 and N. Now since we have an action of SOr(C) on this ﬁber, we can restrict ourselves to elements N+1 ˜ N+1 A(t) ∈ Mr such that A0 = Ir, we denote such locus by Mr . Now one can show, after d some computations, that the Jacobian matrix associated to (fij)i,j,d

∂f d ! Jac((f d ) ) = ij ij i,j,d ∂Xm kl (i,j,d),(k,l,m)

+ ˜ N+1 has in fact a full rank at each point of VN ∩ Mr . Roughly speaking, to see this on can decompose it to a number of sub-matrices using the index i. For example, the first d sub-matrix is the set of lines which contain the partial differentials of the polynomials f1j. d One remarks here that this sub-matrix is the only one which contains the variables X11. So the resulting sub-matrices have full rank, and their lines are linearly independents. It follows by the Jacobian criterion of smoothness that the fiber of α+ over the identity is + smooth. Since this fiber is a fibration over {±1} by the determinant. We deduce that VN is smooth too.

Using the claim and [BL04] Lemma 6.3b), we deduce that Qσ,± are reduced.

+ Example 4.2.6. We give an example in rank 2 of a non trivial element of VN . If N is odd, then one can take 1 + t−N 1 + it−N . −1 + it−N 1 − t−N One can use it to generate a bunch of examples in any rank. 4.2. Loop groups and uniformization theorem 81

4.2.3 Central extension

Consider the central extension of SLr(K )

0 → Gm → SLdr(K ) → SLr(K ) → 0.

± σ± The actions of σ lift to SLdr(K ) giving a central extension of SLr(K )

σ± σ± 0 → Gm → SLdr(K ) → SLr(K ) → 0.

Indeed, let R be a C−algebra, for γ ∈ SLr(R((t))) let a(γ) b(γ) γ = c(γ) d(γ) be its decomposition with respect to R((t)) = VR⊕R[[t]]. Recall that V is a complementary vector subspace of O in K . By [BL94], an element of SLr(R((t))) is given, locally on Spec(R), by a pair (γ, u) where f f γ ∈ SLr(R((t))), u ∈ Aut(VR) such that u ≡ a(γ) mod End (VR), where End (VR) ⊂ End(VR) is the set of ﬁnite rank endomorphisms of VR. By [BL94] Proposition 4.3, the f map γ −→ a(γ) is a group homomorphism from SLr(R((t))) onto the group Aut (VR) of f units of End(VR)/End (VR). It follows that a(γ−1) = a(γ)−1, hence −1 −1 f u ≡ a(γ ) mod End (VR).

So, deﬁne the following actions on SLdr(K ) σ+ :(γ, u) −→ ( tγ(−t)−1, tu(−t)−1),

− t −1 −1 t −1 −1 σ :(γ, u) −→ ( Jr γ(−t) Jr ,Jr u(−t) Jr ). ± Clearly these are involutions which lift σ on SLr(K ).

The Lie algebra attached to SLc r(K ) is given by the central extension

0 → C → slcr(K ) → slr(K ) → 0. (4.2) ˆ It is in fact isomorphic to the aﬃne Lie algebra L(slr) = slr(K ) ⊕ C, with the Lie algebra structure given by dα [(α, u), (β, v)] = [α; β], Res(Tr( β)) , dt where Res stands for the residue. By pulling back the exact sequence (4.2) via the inclusions σ± slr(K ) ,→ slr(K ) we get the central extensions

0 / C / slcr(K ) / slr(K ) / 0 O O

σ± σ± 0 / C / slcr(K ) / slr(K ) / 0,

± where σ act on sclr(K ) by their actions on the ﬁrst summand (which are given in Lemma 4.2.7 below). These are (after adding scaling elements) aﬃne Kac-Moody Lie algebras of (2) σ± twisted type Ar−1. They are in fact the Lie algebras of the twisted groups SLc r(K ) . 82 Chapter 4. Conformal Blocks

σ± Lemma 4.2.7. The Lie algebras associated to SLc r(K ) are the twisted aﬃne Lie algebras of slr(K ) given by ± σ± Lb(slr, σ ) = slr(K ) ⊕ C, ± where the actions of σ on slr(K ) are given by

σ+(g(t)) = − tg(−t),

− t −1 σ (g(t)) = −Jr g(−t)Jr . Proof. The proof is straightforward, we just remark that

t −1 t (Ir + εα) = Ir − ε α, where ε2 = 0.

4.3 Determinant and Pfaﬃan line bundles

Let T be a locally noetherian C−scheme. Denote by p1 and p2 the projection maps from X × T to X and T respectively. Let E be a vector bundle over X × T . The derived direct image complex Rp2∗(E ) is represented by a complex of vector bundles 0 → F0 → F1 → 0. −1 The line bundle DE := det(F0) ⊗ det(F1) over T is independent of the choice of the representing complex and is called the determinant of cohomology of E . The determinant of the universal family L over X ×SU X (r) is called the determinant bundle over SU X (r). Let κ be a σ−invariant theta characteristic over X.

Proposition 4.3.1. Let (E , ψ) be a family of σ−alternating vector bundles over X pa- ∗ rameterized by T , with a σ−alternating non-degenerated form ψ : E ⊗ σ E −→ OX×T . Let ∗ Eκ = E ⊗ p1κ. Then the determinant of cohomology line bundle DEκ admits a square root

PEκ which we call Pfaﬃan of cohomology line bundle.

Proof. Consider the family π∗Eκ over Y . It is equipped with a non-degenerated quadratic form with values in KY . Indeed, by the projection formula, ψ induces an isomorphism

∼ ∗ π∗Eκ = π∗(σ Eκ) ∼ ∗ −1 ∗ = π∗(Eκ (q1 (R))) ⊗ q1KY ∼ ∗ ∗ = (π∗Eκ) ⊗ q1KY , where the last isomorphism is the relative duality (see [Har77] Ex III.6.10) and q1 : Y ×T → Y is the ﬁrst projection. In fact the associated bilinear form is given by the composition

∗ ∗ ∗ ∗ π∗Eκ ⊗ π∗Eκ −→ π∗(p1KX ) = q1(KY ⊗ ∆) ⊕ q1KY −→ q1KY .

Since we project on the −1 eigenspace of the linearization on KX (recall that π∗KX = KY ∆⊕KY ) and because ψ is σ−alternating, we deduce that this bilinear form is symmetric.

We can apply now [LS97] Proposition 7.9 to get a square root of Dπ∗Eκ . To ﬁnish the proof we just have to remark that

DEκ = Dπ∗Eκ . 4.3. Determinant and Pfaﬃan line bundles 83

σ,− In particular, if we consider the universal family over X × SU X (r), we get, for each σ−invariant theta characteristic κ, a Pfaﬃan of cohomology line bundle Pκ over σ,− SU X (r). On the other hand, consider the character χ : SLdr(O) → Gm which is just the second projection (recall that SLdr(O) splits). More precisely, a point of SLdr(O) can be represented locally on Spec(R) by a pair (γ, u), for γ ∈ SLr(R[[t]]) and u an automorphism of VR such f −1 that a(γ) ≡ u mod End (VR). So χ sends this point to det(a(γ) u). To this character one may associate a line bundle Lχ over Q (see [BL94] §3). Moreover Lχ is isomorphic to the pullback of the determinant bundle.

σ− Lemma 4.3.2. The restriction of the character χ to SLdr(O) has a square root which we denote by χ−.

σ− Proof. With the notations of the proof of Proposition 4.2.4, one can see that SLr(O) is 0 the direct limit of the parabolic subgroups PN . So just take the direct limit in Proposition 4.2.4.

σ,− Let L− be the line bundle over Q deﬁned by the character χ− and denote by q : σ,± σ,± Q −→ SU X (r) the quotient maps (there should be no confusion about which map is considered). σ,+ Denote by D the determinant of cohomology line bundle over SU X (r) and by L its pullback to Qσ,+.

σ,− Lemma 4.3.3. The pullback of the Pfaﬃan line bundles Pκ to Q are independent of κ and they are all isomorphic to L−.

σ,− N Proof. It is known (see [Hei10]) that Pic(Q ) = Z , for some integer N. Since all the Pκ are the square roots of the same line bundle D, the result follows.

σ,+ Remark that the line bundle D over SU X (r) does not admit a square root in the ramiﬁed case. This can be seen using Theorem 3.2.17 and the fact that P ∩ Q is not principally polarized (see Theorem 3.2.17 for the notations).

If the cover π : X → Y is étale, then for each σ−invariant theta characteristic the σ,+ universal family over SU X (r) is equipped with a non-degenerated quadratic form with values in KX (this can be seen as in the proof of Proposition 4.3.1). Hence its determinant of cohomology admits a square root, a Pfaffian of cohomology bundle. Moreover, using the functorial definition, one can show that the determinant bundle over σ,+ the stack UX (r) has also a square root attached to any σ−invariant theta characteristic. We keep denoting these line bundles by Pκ, and this should produce no confusion since we always explicitly mention the considered stack.

Lemma 4.3.4. Assume that π is ´etale.Let κ be an even σ−invariant theta characteristic σ,+ σ,+ on X, then the Pfaﬃan line bundle Pκ over UX,0 (r) descends to the moduli space UX,0 (r), σ,+ σ,+ where UX,0 (r) is the connected component of UX (r) of vector bundles with determinant in the connected Prym variety P0 of π : X → Y . Proof. Let κ be a σ−invariant theta characteristic on X and a = (E, q, ψ) be a point of σ Quot (C) (see subsection 2.4.3 for the notations). We assume that E is stable, then the stabilizer of a under the action of SL(H) is just {±1}. The action of this stabilizer on 84 Chapter 4. Conformal Blocks

h1(E⊗κ) σ,+ (Pκ)a is by definition multiplication by g , for g ∈ {±1}. Since UX,0 (r) is connected, we have ( 1 if r ≡ h0(κ) ≡ 1 mod 2 h1(E ⊗ κ) = 0 otherwise. This can be seen using Theorem 3.2.13. Since κ is even, it follows that −1 acts trivially on (Pκ)a, for any a. Using Kempf’s Lemma we deduce the result. Now we show the existence of the Pfaffian divisor. Assume that π : X → Y is étale. For a theta characteristic κ over X we denote by Θκ the divisor in UX (r, 0) supported on vector bundles E such that E ⊗ κ has a non-zero global section.

∗ Lemma 4.3.5. There exists a theta characteristic κ0 over Y , such that, if κ = π κ0, then σ,+ the restriction of the divisor Θκ ⊂ UX (r, 0) to UX,0 (r) is again a divisor. Moreover there σ,± exists an eﬀective divisor Ξκ in UX (r) such that O(Ξκ) = Pκ and

2Ξκ = Θκ.

Proof. Using the Hitchin system (Precisely Theorem 3.2.13) we can reduce the question to ∗ r = 1. Now let κ0 be a theta characteristic on Y such that the restriction of θκ = Tκ θ to the (connected) Prym variety P is a divisor, where θ ⊂ PicgX −1(X) is the Riemann theta 0 gX −1 ∗ divisor and Tκ : Pic (X) → Pic (X) is the translation map by κ = π κ0. Then we see easily that such κ gives the result. To construct such κ0 we proceed as follows: Consider the direct image map P → UY (2, ∆) from P to the moduli space of semistable rank two ˜ ˜ vector bundles over Y of determinant ∆. Then consider the linear system |θ + θ∆|, where ˜ gY −1 ˜ θ ⊂ Pic (Y ) is the Riemann theta divisor and θ∆ is its translation by ∆. Then there ˜ ˜ is a canonical morphism ϕ : UY (2, ∆) → |θ + θ∆|, which sends a vector bundle E to the gY −1 0 ∼ divisor D(E) = {L ∈ Pic (Y )|h (E ⊗L) > 1}. Since π∗L = π∗L⊗∆ for any L ∈ P, one ˜ ˜ ˜ ˜ remarks that the composition ϕ◦π∗ factorizes through the linear system |θ+θ∆|+ ⊂ |θ+θ∆| defined as the invariant locus with respect to taking the tensor product with ∆. Now it ˜ ˜ is sufficient to take κ0 such that the associated hyperplane in |θ + θ∆| does not contain ˜ ˜ entirely the linear system |θ +θ∆|+, this is possible because the above linear system is base point free. σ,+ Moreover, as in [LS97] §7.10, whenever the restriction of Θκ to UX,0 (r) is a divisor, there is an effective divisor Ξκ such that 2Ξκ = Θκ| σ,+ . In particular, Pκ has a non-zero global UX (r) section.

4.4 Generalized theta functions and conformal blocks

Assume in this section that the cover π : X → Y is ramified. We have formulated the uniformization theorem over a single ramification point. However we can use a bunch of points to uniformize our moduli stack. If we consider all the ramification points R, then we get the following σ,± ∼ Y σ,± σ± SU X (r) = Qp /SLr(AR) , p∈R

σ,± σ± σ± 0 σ,± where Qp = SLr(Op) \SLr(Kp) , and AR = H (X r R, OX ). Of course all Qp are isomorphic, but we emphasis on the ﬁxed points. Roughly speaking, this isomorphism can be seen as follows: choose a formal neighborhood Dp of each p ∈ R. Then giving a σ−symmetric vector bundle (E, ψ) of trivial determinant, we choose a σ−invariant local trivializations ϕp near each p and a σ−invariant trivialization ϕ0 on X rR. Then the corresponding point of the right hand side is just is the class of 4.4. Generalized theta functions and conformal blocks 85

−1 (ϕp ◦ϕ0 )p∈R. Conversely, giving a class of functions (fp)p∈R of the RHS, we can construct a σ−symmetric vector bundle by gluing the trivial bundles on Dp and X r R using the functions fp.

σ,− ∗ We have seen that the line bundle L− over Qp is isomorphic to q Pκ and that the σ,+ ∗ Q σ,± σ,± line bundle L over Qp is isomorphic to q D. For x ∈ R, let qx : p∈R Qp → Qx be the canonical projection. We deﬁne the line bundles

O ∗ O ∗ L− = qpL− and L = qpL p∈R p∈R

Q σ,− Q σ,+ over p∈R Qp and p∈R Qp respectively. One can see that L− and L are in fact the Q σ,± σ,± pullback via the projections p∈R Qp → SU X (r) of the line bundles Pκ and D respec- σ± tively. In particular, both of these line bundles have canonical SL(AR) −linearizations. In fact these are the only ones due to the following

σ± Proposition 4.4.1. SLr(AR) are integral and they have only the trivial character. Proof. The proof is inspired from [LS97]. Q σ,± σ,± Using the local triviality of the projection p∈R Qp → SU X (r) and Proposition 4.2.5 σ± we deduce that SLr(AR) are reduced. σ± Now, since connected ind-groups are irreducible, it is suﬃcient to prove that SLr(AR) is connected. For a points p1, . . . , pk ∈ XrR we denote by Ri = R∪{p1, σ(p1), . . . , pi, σ(pi)}. We claim the following Claim. We have an isomorphism

σ± σ± ∼ σ± SLr(ARi ) /SLr(ARi−1 ) = (Qpi × Qσ(pi)) , where the action of σ± on the right hand side is given by σ±(f, g) = (σ±(g), σ±(f)).

σ± σ± Proof. We have a canonical map SLr(ARi ) → (Qpi ×Qσ(pi)) which is clearly trivial on σ± σ± σ± σ± SLr(ARi−1 ) . Hence we deduce a map SLr(ARi ) /SLr(ARi−1 ) → (Qpi × Qσ(pi)) which is actually injective. Now, by considering the uniformization over the two points {pi, σ(pi)}, we get

σ,± ∼ σ± σ± SU X (r) = (Qpi × Qσ(pi)) /SLr(A{pi,σ(pi)}) .

σ± Hence, for an C−algebra S, giving a point of (Qpi × Qσ(pi)) (S) is the same as giving an anti-invariant (σ−symmetric or σ−alternating following ±) vector bundle E over XS ∗ r ∗ and a trivialization δ : E| ∗ → X × , where X = X {p , σ(p )}. For an S−algebra XS S C S S r i i 0 0 ± S , let T (S ) be the space of σ −invariant trivializations of ES0 over XS,i−1 = XS r Ri−1. σ± Then SLr(ARi−1 ) acts on T , and in fact it is a bundle under that group. Moreover δ ˜ σ± −1 induces a map δ : T → SLr(ARi ) by sending a trivialization φ to φ ◦ δ . Associating to (E, δ) the map δ˜ gives an inverse to the above inclusion.

σ± ∼ It is clear to see that (Qpi × Qσ(pi)) = Qpi = SLr(Opi )\SL(Kpi ) which is simply connected. So using the homotopy exact sequence, we deduce that

π0(SLr(ARi )) = π0(SLr(ARi−1 )).

σ± σ± Now let g ∈ SLr(AR) and consider g as an element of SLr(K) , where K is the function ﬁeld of X. By [Tit79] (see also [PR08a] Section 4), we know that the special unitary groups 86 Chapter 4. Conformal Blocks are simply connected and quasi-split. Steinberg ([Ste62]) has showed the Kneser-Tits property for quasi-split simply connected groups over any ﬁeld (Recall that this property means that these groups are generated by the unipotent radicals of their standard parabolic sub- σ± Q groups). So applying that to SLr(KX ) , we can assume that g = i exp(Ni), where Ni σ± 1 are nilpotent elements of slr(KX ) . Let {p1, . . . , pk} be the poles of Ni. For t ∈ A , we Q 1 σ± let gt = i exp(tNi). Then for any t ∈ A we see that gt ∈ SLr(ARk ) and t → gt is a σ± σ± path in SLr(ARk ) that relates g to the identity. Hence SLr(ARk ) is connected. So σ± the same is true for SLr(AR) by what we have shown above.

σ± Now let λ be a character of SLr(AR) , seeing λ as a function, we consider its derivative σ± at the identity which turns out to be a Lie algebras morphism from slr(AR) to the trivial σ± algebra C. However, the aﬃne algebra slr(AR) equals the direct sum of two commutator subalgebras. Indeed, the algebra slr(AR) equals to its commutator, and we have eigenspace decomposition with respect to σ±

slr(AR) = g−1 ⊕ g1, it follows

[g−1 ⊕ g1, g−1 ⊕ g1] = [g−1, g−1] ⊕ [g−1, g1] ⊕ [g1, g−1] ⊕ [g1, g1].

σ± Hence slr(AR) = g1 = [g−1, g−1] ⊕ [g1, g1]. So the derivative of λ at the identity is zero. Since λ is a group homomorphism, its derivative is identically zero everywhere. Since σ± SLr(AX ) is integral, we can write it as limit of integral varieties Vn and for n large

1 ∈ Vn, so λ|Vn = 1, hence λ = 1.

Fix an integer k > 0. For any dominant weight λ± ∈ Pσ,±, there is a line bundle L (λ±) σ,± σ± over Q associated to the principal SLr(O) −bundle:

σ± σ,± SLr(K ) −→ Q ,

−λ± σ± deﬁned using the character e on SLr(O) . Further, it is shown in [Kum87] that the space of global sections of powers of L (λ) is isomorphic to the dual of the irreducible ± ± highest integrable representation of Lb(slr, σ ) associated to λ . ± ± We are mainly interested in the case where λ = λ0 . Denote by H±(k) the highest ± ± weight representation of level k of Lb(slr, σ ) associated to the weight λ0 . It is called the basic representation of level k. So the above result of [Kum87] (see also [Mat88]) can be formulated as follows

0 σ,− ∗ k Theorem 4.4.2 (Kumar, Mathieu). 1. The space H (Q , q Pκ ) is canonically iso- − morphic, as Lb(slr, σ )−module, to the dual of the basic representation H−(k).

0 σ,+ ∗ k + 2. The space H (Q , q D ) is canonically isomorphic, as Lb(slr, σ )−module, to the dual of the basic representation H+(k).

+ − Note that by Remark 4.1.1, when r is even, the weight λ0 has level 2 while λ0 is of level 1. This explains why we have to take the determinant line bundle in σ+ case and the Pfaﬃan line bundle in σ− case.

The point that should be stressed here is that in [Kum87], Kumar has deﬁned the ind- group SLr(O)\SLr(K ) using representation theory of Kac-Moody algebras. It is shown in [BL94] that this construction coincides with the usual functorial deﬁnition. Moreover, 4.5. Application: An analogue of a result of Beauville-Narasimhan-Ramanan 87

Pappas and Rapoport have claimed in [PR08a] (page 3) that the constructions of Kumar coincide with their definitions of the Schubert varieties. In particular, we deduce in our spacial case that the ind-variety structure on the twisted flag varieties Qσ,± are the same as those defined by Kumar.

Using the above results and assumptions, we deduce the following result which has been conjectured in a more general context by Pappas and Rapoport ([PR08a] Conjecture 3.7).

Proposition 4.4.3. We have isomorphisms

σ−  slr(AR) 0 σ,− k ∼ Y 0 σ,− k H (SU X (r), Pκ ) =  H (Qp , L−) , p∈R

σ+  slr(AR) 0 σ,+ k ∼ Y 0 σ,+ k H (SU X (r), D ) =  H (Qp , L ) . p∈R

σ± σ± Proof. Since SLr(AR) and Q are integral, the result follows, using the K¨unnethfor- mula, from [BL04] Proposition 7.4.

Now, Lemma 4.4.3 and Theorem 4.4.2 imply our main result

Theorem 4.4.4. Let k > 2, we have

0 σ,− k 1. The space of global sections H (SU X (r), Pκ ) is canonically isomorphic to the conformal block space Vσ,−(k).

0 σ,+ k 2. The space of global sections H (SU X (r), D ) is canonically isomorphic to the conformal block space Vσ,+(k).

4.5 Application: An analogue of a result of Beauville-Narasimhan- Ramanan

Assume that π : X → Y is étale. Recall that we have constructed in section 3.2a dominant rational map 0 σ,+ q∗ : P 99K UX,0 (r), where P0 is some translate of the connected Prym variety of some étaledouble cover and σ,+ UX,0 (r) is the connected component of the locus of stable σ−symmetric vector bundles E such that det(E) ∈ P, where P is the (connected) Prym variety of X → Y . We fix an 0 identification of P with the Prym variety P of the spectral cover X˜s → Y˜s for general spectral data s ∈ W σ,+. By Lemma 4.3.4, for even theta characteristic κ, the Pfaffian line bundle Pκ descends to σ,+ the moduli space UX,0 (r). We denote it also by Pκ. Lemma 4.5.1. There exists a theta characteristic κ on X such that

0 σ,+ dim(H (UX,0 , Pκ)) = 1. 88 Chapter 4. Conformal Blocks

σ,+ Proof. Let q : X˜s → X be a smooth spectral curve over X attached to a general s ∈ W (see section 3.2 for more details and notations). Let ν = g − 1. First, the pullback of X˜s ν ˜ the determinant bundle via q∗ : J → UX (r, r(gX − 1)) is the line bundle O(θ) attached X˜s to the Riemann theta divisor θ˜ over J ν . Let S ⊂ P0 ⊂ J ν be the locus of line bundles X˜s X˜s L such that q∗L is semi-stable. By [BNR89] Proposition 5.1, the complement of the locus ν of line bundles L ∈ J such that q∗L is semi-stable is contained properly in θ. Since the X˜s restriction of θ to P0 is still a divisor which is actually ample, we deduce (as in [BNR89] Proposition 5.1, b)) that the codimension of the complement of S in P0 is at least 2. Since σ,+ q∗ : S → UX,0 (r) is dominant (note that we need to make a translation by certain line bundle after taking the direct image in order to get degree 0 vector bundles), we deduce an injection 0 σ,+ 0 0 ˜ H (UX,0 (r), Pκ) ,→ H (P , ξ), ˜ 0 0 σ,+ where ξ is a line bundle deﬁning a principal polarization on P . So h (UX,0 (r), Pκ) is at most 1. But by Lemma 4.3.5, we know that for some κ, the line bundle Pκ has a non-zero global section. This ends the proof.

Let ξ and ξ˜ be line bundles deﬁning principal polarizations on P and P respectively. σ,+ σ,+ We also denote by Pκ the restriction of Pκ to SU X (r) ⊂ UX,0 (r). Theorem 4.5.2. We have an isomorphism

0 ∗ ∼ 0 σ,+ H (P, rξ) = H (SU X (r), Pκ). In particular we deduce 0 σ,+ gY −1 dim(H (SU X (r), Pκ)) = r . Proof. Consider the following commutative diagram

P0 ∩ Q0 × P / P

σ,+ σ,+ SU X (r) × P / UX,0 (r),

0 where Q is some translation of the Prym variety of X˜s → X. Using [BNR89] Theorem σ,+ 3, we deduce that the pullback of the line bundle Pκ to SU X (r) × P is of the form ∗ ∗ p1Pκ ⊗ p2O(rξ). 0 0 σ,+ Now the rational map P ∩ Q −→ SU X (r) is dominant (Theorem 3.2.17). It follows, by the same argument used in the proof above, that the map

0 σ 0 0 0 ˜ H (SU X (r), Pκ) → H (P ∩ Q , ξ) is injective, where here we denote abusively by ξ˜ the restriction of ξ˜ to P0 ∩ Q0 ⊂ P0. Since the two abelian subvarieties P and P ∩ Q are a complementary pair inside P, we obtain using [BNR89] Proposition 2.4 an isomorphism

H0(P0 ∩ Q0, ξ˜) =∼ H0(P, rξ)∗.

Hence we deduce an injective map

0 σ,+ 0 ∗ H (SU X (r), Pκ) ,→ H (P, rξ) . 4.5. Application: An analogue of a result of Beauville-Narasimhan-Ramanan 89

0 ∗ σ,+ Moreover the group P[r] acts on PH (P, rξ) as well as on SU X (r), hence it acts also 0 σ,+ 0 ∗ on the linear system PH (SU X (r), Pκ). Since the projective representation PH (P, rξ) 0 σ,+ 0 ∗ is irreducible, the map H (SU X (r), Pκ) ,→ H (P, rξ) , which is equivariant for these actions, is necessarily an isomorphism.

Appendix A

Anti-invariant vector bundles via representations

In this appendix, we study the anti-invariant vector bundles as representations of the fundamental group. Narasimhan and Seshadri established a bijection between irreducible unitary representations of π1(X) modulo conjugation and the isomorphism classes of stable vector bundles of degree 0. So what can we say about σ−anti-invariant vector bundles?

We treat the étalecase. So assume that the cover π : X → Y is unramified and denote −1 the genus of Y by g. Fix y0 ∈ Y and let π (y0) = {x1, x2}. Let α1, . . . , αg, β1, . . . , βg be the generators of π1(Y, y0), with the relation Y [αi, βi] = 1. i Denote hereafter by U(r) the complex unitary group of rank r. Suppose further that the −1 line bundle ∆ = det(π∗OX ) is given by the representation ρ∆ : π1(Y, y0) → U(1) defined by αi → 1, βj → 1, for i = 1, . . . , g, j = 2, . . . , g,

β1 −→ −1.

With these assumptions, one can see that the map π∗ : π1(X, x1) −→ π1(Y, y0) is injective and its image is equal to the kernel of ρ∆. Hence it is generated by

2 α1, . . . , αg, β1 , β2, . . . , βg,

−1 −1 −1 −1 β1α2β1 , . . . , β1αgβ1 , β1β2β1 , . . . , β1βgβ1 . Note that one can easily check that the relation between the commutators of these generators is satisﬁed. Note also that π1(X, x2) has the same image in π1(Y, y0) as π1(X, x1). In fact σ induces ∗ ∼ an isomorphism σ : π1(X, x1) = π1(X, x2). Moreover if γ is the unique path (up to homotopy equivalence) on X that lifts β1 and starting from x2 (it is a path from x2 to x1), then ∼ conjugation by γ induces an isomorphism Cγ : π1(X, x1) −→ π1(X, x2). The composition of these two maps Cγ σ∗ ϑ : π1(X, x1) −→ π1(X, x2) −→ π1(X, x1) 2 gives an automorphism of the group π1(X, x1) whose square ϑ is an inner automorphism, 2 namely it is a conjugation by β1 if identify π1(X, x1) with its image in π1(Y, y0).

Now consider the case of line bundles. We have the following result 92 Appendix A. Anti-invariant vector bundles via representations

Proposition A.0.1. A line bundle L over X is anti-invariant if and only if the associated representation ρL : π1(X, x1) −→ U(1) is ϑ−equivariant, i.e. for all α ∈ π1(X, x1) we have

−1 ρL(ϑ(α)) = ρL(α) .

∗ −1 −1 Proof. The representation associated to σ L and L are ρL◦ϑ and ρL respectively. Hence we have an equivalence

∗ ∼ −1 −1 σ L = L ⇔ ρL(ϑ(α)) ≡ ρL(α) .

But, since U(1) is abelian, two representation are conjugate if and only if they are equal.

Using this Proposition, we deduce

Corollary A.0.2. The locus of anti-invariant line bundles has 4 connected components.

Proof. Remark that the automorphism ϑ induces an automorphism on the image of π1(X, x1) in π1(Y, y0) given by conjugation by β1. Hence if ρ is an ϑ−equivariant representation, then we have −1 −1 −1 −1 ρ(β1αiβ1 ) = ρ(αi) , ρ(β1βjβ1 ) = ρ(βj) . 2 2 In particular, since ϑ(β1 ) = β1 , we have

2 2 −1 ρ(β1 ) = ρ(β1 ) ,

2 hence ρ(β1 ) ∈ {±1}. Moreover the equality

g −1 −1 Y α1β1α1 β1 [αi, βi] = 1, i=2 implies g −1 Y β1α1β1 = [αi, βi]α1. i=2 −1 Since U(1) is abelian, it follows that ρ(β1α1β1 ) = ρ(α1). So we deduce that ρ(α1) = −1 2 ρ(α1) , hence ρ(α1) ∈ {±1}. So the values of ρ at α1 and β1 classify the connected components.

Consider now the case of rank r vector bundles. Recall that by the theorem of Narasimhan-Seshadri, any stable vector bundle over X of rank r is uniquely associated to an equivalence class of irreducible unitary representation of π1(X, x1). It is quite easy to see that if the representation ρE associated to E is ϑ−equivariant, then E is an anti- invariant vector bundle. In fact we have an equivalence

Theorem A.0.3. Let E be a stable vector bundle and ρE its associated representation. Then E is σ−anti-invariant if and only if ρE is ϑ−equivariant, i.e for any α ∈ π1(X, x1), ρE veriﬁes t −1 ρE(ϑ(α)) = ρE(α) . In particular, we deduce the locus of stable anti-invariant vector bundles has 4 connected components. Appendix A. Anti-invariant vector bundles via representations 93

Proof. Let Repϑ(r) be the space of ϑ−equivariant irreducible unitary representations of π1(X, x1). The real orthogonal group O(r, R) (which equals the intersection of the complex one with the unitary group) acts on this space and two ϑ−equivariant representations are equivalent if and only if they are equivalent modulo a conjugation by an element of O(r, R) (here use the fact that the action by conjugation of U(r) on Repϑ(r) is free modulo the center of U(r)). From what we have said above we deduce an injective map

ϑ σ Rep (r)/O(r, R) −→ UX (r),

σ σ,+ where UX (r) is the locus of stable σ−anti-invariant vector bundles, i.e. the union UX (r)∪ σ,− UX (r). Now we can calculate the dimension of the left hand side. Giving a ϑ−equivariant 2 representation is the same as giving elements ρ(α1), . . . , ρ(αg), ρ(β1 )ρ(β2), . . . , ρ(βg) ∈ U(r) subject to the conditions

g t Y t 2 −1 2 ρ(α1) ρ(α1) [ρ(αi), ρ(βi)] = Ir, ρ(β1 ) = ρ(β1 ). i=2

2 2 2 r(r−1) So ρ(β1 ) ∈ O(r, R). Hence this amounts to a real dimension equals (2g − 1)r − r + 2 . r(r−1) Since we take the equivalence classes modulo O(r, R), we have to subtract 2 . Thus the complex dimension is given by

ϑ 2 σ,+ dimC(Rep ) = r (g − 1) = dim(UX (r)). So we deduce that we have an isomorphism

ϑ ∼ σ Rep (r)/O(r, R) = UX (r).

σ Now we deduce that UX (r) has 4 connected components parameterized by the determi- 2 ϑ nant of the values of the representation ρ at α1 and β1 . Indeed, let ρ ∈ Rep (r), then from 2 the conditions above we deduce that the det(ρ(α1)) = ±1 and det(ρ(β1 )) = ±1. These two equalities gives the required parameter.

Remark A.0.4. Note that, when the rank r is odd, the action of Repϑ(1) on Repϑ(r) by ϑ ∼ 2 ϑ ∼ 2 multiplication induces a transitive action of π0(Rep (1)) = µ2 on π0(Rep (r)) = µ2, where µ2 = {±1}. This is actually what we have proved using the Hitchin systems in Theorem 3.2.24.

Appendix B

Stability of the pullback of stable vector bundles and application

Let π : X → Y be a ramiﬁed double cover, where X and Y are smooth irreducible projective curves. We will show that the pullback of a stable vector bundle over Y by π is stable. We are grateful to George Hitching for showing this result to us. However, we have slightly improved the proof. As an application, we construct examples of stable anti-invariant vector bundles.

Lemma B.0.1. Let E be a vector bundle on Y , and F ⊂ π∗E be a subbundle. Then F descends to Y iﬀ the canonical linearization on π∗E gives a linearization on F : σ∗F −→∼ F .

Proof. The direct implication is clear. So let F ⊂ π∗E such that the canonical linearization φ : σ∗(π∗E) → π∗E restricted to a linearization of F

φ : σ∗F → F.

∗ If p is a ramiﬁcation point then φ = id on (π E)p (by lemma 1.1.1), so its restriction to Fp is also the identity. Thus, again by lemma 1.1.1, F descends to Y .

Theorem B.0.2. Let E → Y be a stable vector bundle. Then π∗E is also stable.

Proof. Let’s denote by

s(E,F ) = deg(E)rk(F ) − deg(F )rk(E), for two vector bundles F ⊂ E. It is clear that E is stable if and only if s(E,F ) > 0 for all proper subbundle F ⊂ E. Note also that s(π∗E, π∗F ) = 2s(E,F ) for any subbundle F ⊂ E over Y . Let F ⊂ π∗E be any proper subbundle. Let P ⊂ π∗E be the bundle generated by F +σ∗F , and N such that 0 → N → F ⊕ σ∗F → P → 0. Now, we claim that both P and N descend to Y . By lemma B.0.1, We have just to prove that the canonical linearizationσ ˜ : σ∗π∗E → π∗E gives a linearization on these bundles. Let F ⊕ σ∗F → π∗E be the canonical map, it is clearly σ−equivariant, so its image P and its kernel N are both σ−invariant. Hence P = π∗P 0 and N = π∗N 0 for some sub-bundles P 0 and N 0 of E. ∗ Let rE and dE (resp. rP , dP , rN , dN , rF , dF ) be the rank and the degree of π E (resp. P , N, F ). We have 96 Appendix B. Stability of the pullback of stable vector bundles and application

∗ ∗ s(π E,P ) + s(π E,N) = rP dE − rEdP + rN dE − rEdN

= (rP + rN )dE − rE(dP + dN )

= 2rF dE − 2rEdF = 2s(π∗E,F ).

But E is stable, so if P 0 or N 0 is proper (non trivial) sub-bundle of E, we deduce s(π∗E,F ) > 0. So suppose that P 0 = E and N 0 = 0, in this case we have

π∗E =∼ F ⊕ σ∗F, as π is ramiﬁed, the vector bundle F ⊕ σ∗F descends to Y if and only if F descends, to see this, choose a ramiﬁcation point p ∈ R, then from the commutative diagram

∼ σ∗F ⊕ F / F ⊕ σ∗F

o o ∼ σ∗π∗E / π∗E, we deduce that over the ramiﬁcation point p the linearization φ is trivial, so it induces a linearization σ∗F → F . Thus by lemma B.0.1, F descends to Y . Let F 0 be the vector bundle on Y such that π∗F 0 = F . As π∗ is injective, we deduce that E =∼ F 0 ⊕ F 0, which contradicts the stability of E.

Remark B.0.3. The last result is completely false if the cover is ´etale,a counterexample can easily be constructed: take any line bundle L on X of degree 1, then its direct image π∗L is rank two vector bundle of degree 1 on Y which is (by a result of Beauville) semi-stable. So it is stable. But we have ∗ ∼ ∗ π π∗L = L ⊕ σ L, which is not stable.

Proposition B.0.4. Let F be a stable orthogonal (resp. symplectic) vector bundle on Y , then E = π∗F is a stable σ−anti-invariant vector bundle on X with a σ−symmetric (resp. σ−alternating) isomorphism ψ : σ∗E −→∼ E∗.

Proof. Let φ : σ∗E −→∼ E be the canonical linearization, which is the identity over the ramiﬁcation points. Let ξ : F −→∼ F ∗, the orthogonal (resp. symplectic) isomorphism. Denote ψ = π∗ξ ◦ φ. We have to prove that ψ is σ−symmetric (resp. σ−alternating), i.e.

σ∗ψ = tψ (resp. σ∗ψ = − tψ).

First we have a commutative diagram

φ σ∗E / E

σ∗(π∗ξ) π∗ξ t(σ∗φ) σ∗E∗ / E∗, Appendix B. Stability of the pullback of stable vector bundles and application 97 which is easy to verify. Hence we have

t(σ∗ψ) = t(σ∗(π∗ξ) ◦ σ∗φ) = t(σ∗φ) ◦ t(σ∗(π∗ξ)) = t(σ∗φ) ◦ σ∗(π∗(tξ)) = ± t(σ∗φ) ◦ σ∗(π∗ξ) = ±(π∗ξ) ◦ φ = ±ψ.

Appendix C

On the codimension of non very stable rank 2 vector bundles

In this appendix, we prove directly, without using the result of Laumon [Lau88], that the locus of stable and non very stable σ−symmetric anti-invariant vector bundles of rank 2 has codimension at least 1. Let E be such vector bundle. Denote by ψ : σ∗E −→∼ E∗ and

0 ∗ φ ∈ H (X,E ⊗ σ E ⊗ KX )+

0 be a nilpotent Higgs ﬁeld. In particular φ ∈ H (X, End0(E) ⊗ KX )+. So we can suppose ∼ −1 that det(E) = OX . Then let L be the kernel of φ : E → E ⊗ KX , which is a line bundle of degree −d < 0. We have the following commutative diagram

p 0 / L−1 / E / L / 0

0 φ −1 i 0 / L ⊗ KX / E ⊗ KX / L ⊗ KX / 0.

0 This implies that φ factorizes through L, that’s to say, there exists φ : L → E ⊗ KX such 0 that φ = φ ◦ p. Moreover, as (φ ⊗ 1KX ) ◦ φ = 0, this implies that 0 (φ ⊗ 1KX ) ◦ φ = 0, 0 −1 0 −2 hence φ factorizes through Ker(φ⊗1KX ) = L ⊗KX . Hence there exists s ∈ H (X,L ⊗ KX ) such that φ = i ◦ s ◦ p.

In particular d 6 gX − 1.

But, since φ is invariant, the following diagram

σ∗φ : σ∗E / σ∗E ⊗ σ∗K O X ψ−1 ψ⊗η

t ∗ ∗ φ ⊗ 1KX : E / E ⊗ KX , commutes. It follows that L−1 is ψ isotropic, i.e. ψ induces the zero map σ∗L−1 → L. Furthermore, we have the diagram

p 0 / σ∗L−1 / σ∗E / σ∗L / 0

ψ i∗ 0 / L−1 / E∗ / L / 0, 100 Appendix C. On the codimension of non very stable rank 2 vector bundles which implies that ψ induces a non-zero map σ∗L−1 → L−1, hence it is an isomorphism. Thus L−1 (and so L) is σ−invariant line bundle. Finally, giving a stable and non very stable vector bundle E of rank 2 with a non- zero nilpotent Higgs ﬁeld φ, is the same as giving a σ−invariant line bundle L and an 1 −1 ∗ anti-invariant extension ξ ∈ Ext (L, L )−/C (modulo a scalar) such that deg(L) = d ∈ {1, ··· , gX − 1}, with a non-zero section

0 −2 s ∈ H (X,L ⊗ KX )+.

Fix the degree d. And for simplicity, we suppose that L descends to M ∈ Picd/2(Y ). Deﬁne

d/2 0 −2 Θd = {[M] ∈ Pic (Y ) | h (M KY ∆) > 0}.

The dimension of Θd is given by

min{gY , −d + 2(gY − 1) + n} =: θ(d),

1 −1 ∼ 1 −2 and the dimension of Ext (L, L )− = H (X,L )− can be calculated easily using the fact that the ﬁxed linearization on KX is the positive one. We obtain

1 −1 ∗ 1 −2 dim(Ext (L, L )−/C ) = h (Y,M ) − 1 = d + gY − 2.

Thus, the locus of such bundles has a dimension

0 −2 θ(d) + d + gY − 2 + h (Y,M ⊗ KY ⊗ ∆),

0 −2 but for general M ∈ Θd, h (M ⊗ KY ⊗ ∆) = min{1, d − (gY − 2) − n}. So it follows that the dimension we are looking for is

θ(d) + d + gY − 2 + min{1, d − (gY − 2) − n} = 3(gY − 1) + n

σ,+ = dim(SU X (r)). ∗ σ,+ But this locus inside T SU X (2) is conic, that’s to say stable by the canonical action of ∗ σ,+ C , hence over each point of SU X (2) the ﬁber of this locus has dimension at least 1. This σ,+ implies that its image in SU X (2) has codimension at least 1. 101

Appendix D

Some Results on anti-invariant vector bundles

D.1 Anti-invariance of elementary transformations

Let (E, ψ) be an anti-invariant vector bundle, and ﬁx a ramiﬁcation point p ∈ R. After taking two elementary transformations (negative and than positive)

0 → F → E → Cp → 0, 0 0 → F → E → Cp → 0, we ask if E0 is σ−anti-invariant? First, the restriction of ψ to F gives a map σ∗F → F ∗. Its co-rank at p is either 1 (for general transformation) or 2, and it is 2 if and only if

l ⊂ Q∗,

∗ ∗ ∗ ∗ −1 where l = ker(Ep → Fp ), and Q = {φ ∈ Ep |φ(ψ (φ)) = 0} is the quadric associated to ψ−1. Indeed, we have the diagram

i ∼ ∗ ∗ Fp −→ Ep −→ Ep → Fp .

In fact, the restriction of ψp to Fp is the composition of these maps. So the composed map is of co-rank 2 if and only if l ⊂ Im(ψp ◦ i). But by deﬁnition, the elements of l are those which vanish on Im(i). It follows

∗ l ⊂ Im(ψp ◦ i) =⇒ l ⊂ Q .

Conversely , if we take φ ∈ l which is non-zero, so we have Im(i) ⊂ Ker(φ), but this two spaces are both lines, so Im(i) = ker(φ), therefore

−1 −1 φ(ψp (φ)) = 0 ⇒ ψp (φ) ∈ Im(i) ⇒ φ ∈ Im(ψp ◦ i).

0 Now, taking the ﬁrst transformation such that the co-rank of ψp is 2, assuming that E is stable (which is the general case), and looking to the diagram

0 / σ∗F / σ∗E0

E0∗ /) F ∗ / 0. We see that σ∗F → F ∗ induces a map σ∗E0 → E0∗ if and only if

0 ∗ ker(Fp → Ep) ⊂ ker(Fp → Fp ), 102 Appendix D. Some Results on anti-invariant vector bundles and ∗ 0∗ ∗ Im(Fp → Fp ) ⊂ Im(Ep → Fp ). t This last two conditions are the same (we use here the fact that ψp = ±ψp). So by choosing a line 0 ∗ L = ker(Fp → Ep) ⊂ ker(Fp → Fp ), we get a σ−anti-invariant E0. Moreover, all the elementary transformations of this kind ∗ 1 are parameterized by Pker(Fp → Fp ) = P .

D.2 Another description of anti-invariant vector bundles

Lemma D.2.1. Giving a σ−symmetric anti-invariant vector bundle (E, ψ) over X is the same as giving a pair (E, ϕ), where ϕ is a OY -bilinear perfect form

ϕ : π∗E × π∗E → π∗OX ,

∗ ∗ such that ϕ(a·u, v) = ϕ(u, σ (a)·v) and ϕ(u, v) = σ (ϕ(v, u)) for all u, v ∈ π∗E, a ∈ π∗OX Proof. Assume that we have (E, ψ), then we take

ϕ = π∗ψ,˜

˜ ∗ where ψ : E ⊗ σ E −→ OX is the σ−bilinear forms deﬁned by ψ. We can easily check that ϕ veriﬁes the conditions above. Conversely, suppose that (E, ϕ) is given. First, consider the exact sequence

∗ ∗ 0 → π π∗E → E ⊕ σ E → E|R → 0, where, as always, R denote the ramiﬁcation divisor. So generically we have a canonical ∗ ∗ ∗ isomorphism between π π∗E and E ⊕ σ E. Notice that π ϕ is OX −bilinear form, which ∗ ∼ ∗ veriﬁes for all a ∈ π π∗OX = OX ⊕ σ OX , which is of the form a = a+ + a−, we have

π∗ϕ(a · u, v) = π∗ϕ(u, σ∗(a) · v),

∗ ∗ ∗ where σ (a) = a+ − a−, a · u = au+ + σ (a)u− (we decompose u according to E ⊕ σ E) Taking a+ = 0, u = u+ and v = v+ (resp. u = u−, v = v−) we get

∗ ∗ ∗ ∗ a−π ϕ(u, v) = π ϕ(a−u, v) = π ϕ(u, −a−v) = −a−π ϕ(u, v).

So π∗ϕ(u, v) = 0 for all (u, v) ∈ E × E (resp. (u, v) ∈ σ∗E × σ∗E). ˜ ∗ Hence, we take ψ = π ϕ|E⊗σ∗E. Notice that

0 σ∗(tψ˜) π∗ϕ = ψ˜ 0 according to the decomposition (E ⊕ σ∗E) ⊗ (E ⊕ σ∗E) D.3. Equality between two canonical maps 103

D.3 Equality between two canonical maps

Studying the dominance of the Hitchin map, we had been led to consider the surjec- tivity of its differential. It turns out that it has a canonical identification with the dual of the differential of the pushforward map. Although that this has not been used in this dissertation, we would like to mention it here.

L 0 i ˜ Let s = (si) ∈ W = H (X,KX ) and q : Xs → X the associated spectral curve, let L be a line bundle over X˜s such that E := q∗L is stable vector bundle of rank r and degree 0. Denote by ϕ : E → E ⊗ KX the associated Higgs ﬁeld. Consider the Hitchin map

0 ∗ H : H (X,E ⊗ E ⊗ KX ) −→ W, whose ith component is deﬁned by

i Hi(φ) = T r(φ ).

Theorem D.3.1. With the above assumptions, the differential of the Hitchin map dϕH ∗ is canonically identified with (dL(q∗)) . Proof. First by [Bea91], we have the identification

∼ 1 dLq∗ = H (λ), where λ : q O −→ End(q L) is the canonical map coming from the O −module struc- ∗ X˜s ∗ X˜s ture of L. Hence, by Serre duality

∗ ∼ 1 ∗ ∼ 0 ∗ (dLq∗) = H (λ) = H (λ ⊗ idKX ).

Moreover, by developing the formula H (ϕ+εψ) = H (ϕ)+εdϕH (ψ), we deduce easily

i−1 dϕHi(ψ) = iTr(ψ ◦ ϕ ).

On the other hand, the map

−1 −r+1 ∗ λ : OX ⊕ KX ⊕ · · · ⊕ KX −→ E ⊗ E is given explicitly by

r−1 λ(s0, s1, . . . , sr−1) = s0id + ϕ(s1) + ··· + ϕ (sr−1),

−1 ∗ where ϕ is seen as a map KX → E ⊗ E . Hence λ∗(ψ) = Tr(ψ), Tr(ψ ◦ ϕ),..., Tr(ψ ◦ ϕi−1) .

0 ∗ i−1 Thus H (λ ⊗ idKX )(ψ) = Tr(ψ ◦ ϕ ) i So by taking the automorphism of W given by multiplication by (1, 2, . . . , r), we get the identiﬁcation.

105

Appendix E

Rank 2 case

σ,+ In this appendix, we give another proof of the irreducibility of UX (2) and the fact σ,− that UX (2) has two connected components in the ramiﬁed case. Assume that π : X → Y is ramiﬁed. In this case every vector bundle E over X with trivial ∼ ∗ determinant is an Sp2−bundle, that’s E = E with a symplectic form, and it admits a symmetric one if and only if it is polystable. We see in this particular case that σ−invariant bundles are the same as σ−anti-invariant bundles. So let (E, φ) be a stable σ−invariant bundle with trivial determinant. The triviality of the determinant implies that the type of E must be of the form

τ = (0, ··· , 0),

or τ = (A1, ··· ,A2n), with Ai ∈ {+1, −1}, −1 0 ±1 0 where 0 = and ±1 = . 0 1 0 ±1 We have φ q ψ : σ∗E −→ E −→ E∗, where q is a symplectic form. Let ψ = q ◦ φ. It is not diﬃcult to see that if E has a type (0, ··· , 0), then ψ is σ−symmetric, and it is σ−alternating otherwise. σ,+ σ,− 2n−1 In particular, one deduces that SU X (2) is connected, and SU X (2) has 2 connected components. Moreover, we have surjective maps

σ,+ σ,+ SU X (2) × P → UX (2),

σ,− σ,− SU X (2) × P → UX (2). σ,+ This proves the irreducibility of UX (2). σ,− The group P [2] acts naturally on the set of connected components of SU X (r) in the following way: for λ ∈ P [2] of type υ = (ε1, ··· , ε2n), where εi ∈ {±1}, and for a type σ,− τ = (A1, ··· ,A2n) attached to a connected component of SU X (r), we have

υ · τ = (ε1A1, ··· , ε2nA2n).

∗ ∗ 2n−2 Furthermore this action is free modulo π JY [2]. Since card(P [2]/π JY [2]) = 2 , we σ,− deduce that this action has two orbits. It follows that UX (2) has two connected components.

107

Appendix F

Lefschetz Fixed Point Formula

We have used the Lefschetz ﬁxed point theorem in several places in this thesis, so we would like to explicitly write it down in this appendix. We translate this result into our spacial context. Before that let’s introduce some notations.

Let X be a smooth projective curve, and denote by % : X → X an arbitrary automorphism. Since X is smooth, any fixed point of % is simple. Denote the fixed locus of % by S. Consider now a %−invariant vector bundle E of rank r and let ψ : %∗E → E be an isomorphism. The isomorphisms % and ψ induce an automorphism on the spaces Hi(X,E) for i = 0, 1. We denote these automorphism by %i. Define the Lefschetz number attached to this data by

X i L(%, ψ) = (−1) Tr(%i) i

= Tr(%0) − Tr(%1). Theorem F.0.1 ([AB68], Theorem 4.12). With the above notations, we have

X Tr(ψp) L(%, ψ) = , det(1 − dp%) p∈S where dp% : TpX → TpX denote the diﬀerential of % at p ∈ X.

In particular, since we are mainly interested in involutions, we deduce the following corollary.

Corollary F.0.2. Assuming that % is an involution, we get

0 0 1 1 X Tr(ψp) h (X,E)+ − h (X,E)− − h (X,E)+ + h (X,E)− = . det(1 − d%p) p∈S

109

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